# BubbleML: A Multi-Physics Dataset and Benchmarks for Machine Learning

Sheikh Md Shakeel Hassan1† Arthur Feeney1† Akash Dhruv2 Jihoon Kim3
Youngjoon Suh1 Jaiyoung Ryu3 Yoonjin Won1 Aparna Chandramowlishwaran1
1University of California, Irvine 2Argonne National Laboratory 3Korea
University
$\dagger$ Equal contributions
<EMAIL_ADDRESS>
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###### Abstract

In the field of phase change phenomena, the lack of accessible and diverse
datasets suitable for machine learning (ML) training poses a significant
challenge. Existing experimental datasets are often restricted, with limited
availability and sparse ground truth data, impeding our understanding of this
complex multiphysics phenomena. To bridge this gap, we present the BubbleML
Dataset 111https://github.com/HPCForge/BubbleML which leverages physics-driven
simulations to provide accurate ground truth information for various boiling
scenarios, encompassing nucleate pool boiling, flow boiling, and sub-cooled
boiling. This extensive dataset covers a wide range of parameters, including
varying gravity conditions, flow rates, sub-cooling levels, and wall
superheat, comprising 79 simulations. BubbleML is validated against
experimental observations and trends, establishing it as an invaluable
resource for ML research. Furthermore, we showcase its potential to facilitate
exploration of diverse downstream tasks by introducing two benchmarks: (a)
optical flow analysis to capture bubble dynamics, and (b) operator networks
for learning temperature dynamics. The BubbleML dataset and its benchmarks
serve as a catalyst for advancements in ML-driven research on multiphysics
phase change phenomena, enabling the development and comparison of state-of-
the-art techniques and models.

##  1 Introduction

Phase-change phenomena, such as boiling, involve complex multiphysics
processes and dynamics that are not fully understood. The interplay between
bubble dynamics and heat transfer performance during boiling presents
significant challenges in accurately predicting and modeling these heat and
mass transfer processes. Machine learning (ML) offers the potential to
revolutionize this field, enabling data-driven discovery to unravel new
physical insights, develop accurate surrogate and predictive models, optimize
the design of heat transfer systems, and facilitate adaptive real-time
monitoring and control.

The applications of ML in this domain are diverse and impactful. Consider the
context of high-performance computing in data centers, where efficient cooling
is critical. Boiling-based cooling techniques, such as two-phase liquid
cooling, offer enhanced heat dissipation capabilities, ensuring reliable and
optimal operation of power-intensive electronic components such as GPUs.
Moreover, boiling phenomena play a crucial role in complex processes like
nuclear fuel reprocessing, where precise modeling and prediction of boiling
dynamics contribute to the safe and efficient management of nuclear waste. In
the realm of water desalination, boiling processes are integral, playing a
vital role in thermal desalination methods that provide clean drinking water
in water-scarce regions . These advancements in pivotal areas such as thermal
management, energy efficiency, and heat transfer applications, driven by ML
techniques have far-reaching implications, empowering us to design more
sustainable energy systems, enhance environmental preservation efforts, and
advance engineering capabilities across various domains.

Figure 1: BubbleML Dataset. Capturing diverse two-phase boiling phenomena with
ground truth for key physical variables–velocity, temperature, and pressure.
(a) Single bubble rising from a nucleation site on the heater surface. (b)
Chaotic multi-bubble dynamics—merging, splitting. (c) Flow boiling transitions
from bubbly to slug regime with increasing inlet velocity. The velocity and
temperature fields are obtained by solving equations 1a and 1b, while the
pressure field is obtained by solving the Poisson equation which ensures that
continuity is satisfied.

To train data-driven ML algorithms effectively, we need large, diverse, and
accurately labeled datasets. However, obtaining high-fidelity datasets that
encompass a wide range of phase-change phenomena and operating conditions is a
significant challenge. Boiling processes are highly sensitive to factors like
surface properties, pressure, orientation, and working fluid composition [1].
Additionally, the chaotic nature of vapor interactions and occlusions makes
quantifying boiling processes inherently difficult. Specialized experimental
setups, involving instrumentation, sensors, and high-speed visualization
techniques, come with substantial costs, further limiting the availability of
extensive and accurate large-scale experimental data [59]. As a result, only a
few well-funded research laboratories have access to precise ground truth
data, and even then, this data often lacks fidelity and fails to capture
detailed microscale dynamics, such as local bubble-induced turbulence and its
impact on overall heat transfer. This scarcity of high-fidelity datasets poses
challenges in designing accurate ML models for multiphase and phase change
processes. While scientific ML (SciML) approaches can incorporate physical
knowledge and constraints into the training process to reduce some of this
data burden [44], the validation and quantification of uncertainty still rely
on the availability of ground truth data. Therefore, there is an urgent need
for open, diverse, and large-scale datasets to develop robust models and
advance research in multiphysics problems such as phase change phenomena.

Simulations have played a key role as the third pillar of science in
overcoming the inherent challenges faced by experimental studies in various
scientific domains. High-fidelity multiscale data from simulations complement
and enhance experimental measurements. In the field of phase change,
simulations have successfully modeled transport equations for momentum,
energy, and phase transition enabling accurate measurements of velocity,
pressure, and temperature fields around bubbles [51, 27]. As a result,
simulations serve as powerful tools for understanding and quantifying boiling.
However, SciML researchers often setup their own simulations to generate
ground truth solutions for training and testing their models, rather than
relying on shared benchmark datasets. This is even common among major papers:
[44, 30, 34, 26]. While this approach is reasonable for studying specific,
simple partial differential equations (PDEs), real-world applications of PDE
solvers and simulations often involve large-scale systems with complex
multiphase physics and a combination of Dirichlet and Neumann boundary
conditions [58]. These real-world problems require substantial domain
expertise, engineering time, and computational resources. Performing such
simulations independently is impractical or even infeasible for many ML
researchers. This difficulty in dataset generation has led to a drought of
SciML research to study “real-world" physics problems. Previous efforts to
build benchmark datasets have primarily focused on single- and multiphysics
problems with single-phase dynamics [61, 40, 9, 5].

As a response to the aforementioned challenges, we introduce the BubbleML
Dataset 222Through Zenodo, a permanent DOI for the dataset
10.5281/zenodo.8039786, an extensive and innovative collection of data
generated through Flash-X simulations [15]. This dataset encompasses a wide
range of boiling phenomena, including nucleate boiling of single bubbles,
merging bubbles, flow boiling in different configurations, and subcooled
boiling. Figure 1 provides a visual glimpse into the diverse range of physical
phenomena and variables covered by the dataset. To further enhance its
applicability, the dataset covers various gravity conditions ranging from
earth gravity to gravity at the International Space Station, different heater
temperatures and also different inlet velocities. In total, we present around
80 simulations, each capturing a specific combination of parameters and
conditions. In summary, the key contributions are as follows:

Multiphase and Multiphysics Dataset. A comprehensive dataset encompassing a
wide range of two-phase (liquid-vapor) phase change phenomena in boiling, with
a focus on bubble and flow dynamics. This dataset will will be of great
interest to the scientific machine learning and thermal science communities.

Real-world Validation. Validation against experimental data to ensure the
datasets accuracy and reliability. This validation process enhances the
dataset’s fidelity and establishes a strong connection between simulation and
real-world phenomena.

Diverse Downstream Tasks. BubbleML is designed to facilitate diverse
downstream tasks. To demonstrate the dataset’s potential, we provide two
benchmark scenarios: optical flow for learning bubble dynamics and scientific
machine learning using operator networks to learn temperature dynamics and
estimate heatflux.

##  2 Related Work

Scientific Machine Learning Datasets. There have been several efforts to
develop benchmark datasets for scientific machine learning tasks [61, 40, 9,
5, 56, 23]. Notably, the ERA5 atmospheric reanalysis dataset [23], curated by
the European Center for Medium-Range Weather Forecasting (ECMWF) provides
hourly estimates of a large number of atmospheric, land, and oceanic climate
variables since 1940. It is the most popular publicly available source for
weather forecasting, facilitating the training of neural weather models such
as FourCastNet [41], GraphCast [28], and ClimaX [38]. PDEBench [61] provides
an impressive collection of datasets for 11 PDEs commonly encountered in
computational fluid dynamics. Boundary conditions in scientific simulations
play a crucial role in capturing the dynamics of the underlying physical
systems. The majority of datasets in PDEBench utilize periodic boundary
conditions. Although some datasets encompass Neumann or Dirichlet boundary
conditions, none consider a combination of both which presents a noteworthy
gap in accurately modeling real-world scenarios. Another challenging problem
is the modeling of turbulent Kolmogorov flows and the dataset generated using
JAX-CFD [25] is gaining popularity in benchmarking neural flow models [32,
60]. BlastNet [9] generated using DNS solver, S3D [8] focuses on simulating
the behavior of a single fluid phase solving for compressible fluid dynamics,
combustion, and heat transfer. AirfRANS [5] is a dataset for studying the 2D
incompressible steady-state Reynolds-Averaged Navier–Stokes equations over
airfoils. Current datasets have made commendable strides in addressing single-
and multiphysics scenarios, and provide a valuable foundation for developing
and evaluating SciML algorithms. Nonetheless, their scope falls short in
capturing the range of behaviors and phenomena encountered in phase change
physics.

In contrast, BubbleML focuses on capturing the complex dynamics and physics
associated with multiphase phenomena, particularly in the context of phase
change simulations. Unlike many existing datasets that predominantly utilize a
single type of boundary condition, BubbleML incorporates a combination of
Dirichlet and Neumann boundary conditions [58]. This inclusion enables
researchers to explore and model scenarios where multiple boundary conditions
coexist, enhancing the realism and applicability of the dataset. Moreover, the
presence of “jump” conditions along the liquid-vapor interface adds an
additional layer of complexity. These conditions arise due to surface tension
effects and require careful modeling to accurately capture the interface
behavior [12, 13]. By incorporating such challenges, BubbleML provides a
realistic and demanding testbed for ML models.

Optical Flow Datasets. Optical flow estimation, a classical ill-posed problem
[3] in image processing, has witnessed a shift from traditional methods to
data-driven deep learning approaches.

Middlebury [2] is a dataset with dense ground truth for small displacements,
while KITTI2015 [37] provides sparse ground truth for large displacements in
real-world scenes. MPI-Sintel [7] offers synthetic data with very large
displacements, up to 400 pixels per frame. However, these datasets are
relatively small for training deep neural networks. FlyingChairs [14], a large
synthetic dataset, contains around 22,000 image pairs generated by applying
affine transformations to rendered chairs on random backgrounds.
FlyingThings3D [35] is another large synthetic dataset with approximately
25,000 stereo frames of 3D objects on different backgrounds.

While these datasets have been instrumental in advancing data-driven optical
flow methods, they primarily focus on rigid object motion in visual scenes and
do not address the specific challenges posed by multiphase simulations.
Efforts have been made to capture non-rigid motion in nature, such as piece-
wise rigid motions seen in animals [29]. In boiling, the non-rigid dynamics of
bubbles and the motion of liquid-vapor interfaces play a crucial role in the
distribution and transfer of thermal energy. The BubbleML dataset provides a
unique opportunity to explore and develop optical flow algorithms tailored to
such dynamics. Unlike existing datasets, it offers a diverse range of bubble
behaviors, including merging, growing, splitting, and complex interactions
(see Figure 1). As a result, BubbleML fills a gap by providing challenging
scenarios that involve phase change dynamics. The ability to accurately
predict and forecast bubble dynamics has practical implications in various
fields.

##  3 BubbleML: A Multiphase Multiphysics Dataset for ML

In this section, we start by introducing the preliminary concepts underlying
the SciML learning problem and give insights into the types of simulations and
PDEs involved in this domain. Then, we present an overview of the dataset
pipeline along with its validation against real-world experiments.

###  3.1 Preliminaries

A common application for SciML is approximating the solution of boundary value
problems (BVPs). BVPs are widely used to model various physical phenomena,
including fluid dynamics, heat transfer, electromagnetics, and quantum
mechanics [58, 57, 39, 20]. BVPs take the form: $L[u(x)]=f(x),x\in\Omega$ and
$B[u(x)]=g(x),x\in\partial\Omega$. The goal is to determine the vector-valued
solution function, $u$. $x$ is a point in the domain $\Omega$ and may include
a temporal component. The boundary of the domain is denoted as
$\partial\Omega$. The governing equation is described by the PDE operator $L$,
and the forcing function is denoted as $f$. The _boundary condition_ (BC) is
given by the boundary operator $B$ and the boundary function $g$. $B[u]=g$
ensures the existence and uniqueness of the solution.

There are three common types of BCs: periodic, Dirichlet, and Neumann.
Periodic BCs enforce the equality of the solution at distinct points in the
domain: $u(x_{1})=u(x_{2})$. Dirichlet BCs specify the values of the solution
on the boundary: $u(x)=g(x)$. Neumann BCs enforce constraints on the
derivatives of the solution: $\partial_{n}u(x)=g(x)$ [58]. As seen in Figure
1, BubbleML combines both Dirichlet (no-slip walls, heater, inflow) and
Neumann (outflow) boundaries, which impose constraints on flow and temperature
dynamics. Additionally, the “jump conditions” that govern the transitions
between the liquid and vapor phases use Dirichlet and Neumann boundaries [12].

###  3.2 Overview of PDEs and Flash-X Simulation

A comprehensive description of the simulations is well beyond the scope of
this paper and can be found in [12, 13]. We provide a concise description here
as knowledge of the PDEs is important when training physics-informed models.

The liquid ($l$) and vapor ($v$) phases of a boiling simulation are
characterized by differences in fluid and thermal properties: density, $\rho$;
viscosity, $\mu$; thermal diffusivity, $\alpha$; and thermal conductivity $k$.
The phases are tracked using a level-set function, $\phi$, which is positive
inside the vapor and negative in the liquid. $\phi=0$ provides implicit
representation of the liquid-vapor interface, $\Gamma$ (see Figure 1). The
transport equations are non-dimensionalized and scaled using the values in
liquid and are given as,

$\frac{\partial\vec{u}}{\partial
t}+\vec{u}\boldsymbol{\cdot}\nabla\vec{u}=-\frac{1}{\rho^{\prime}}\nabla
P+\nabla\boldsymbol{\cdot}\Big{[}\frac{\mu^{\prime}}{\rho^{\prime}}\frac{1}{\text{Re}}\nabla\vec{u}\Big{]}+\frac{\vec{g}}{\text{Fr}^{2}}+{\vec{S}^{\Gamma}_{u}}+{S}^{\Gamma}_{P}$
(1a) $\frac{\partial T}{\partial t}+\vec{u}\boldsymbol{\cdot}\nabla
T=\nabla\boldsymbol{\cdot}\Big{[}\frac{\alpha^{\prime}}{\text{Re}\>\text{Pr}}\nabla
T\Big{]}+{S}^{\Gamma}_{T}$ (1b)

where, $\vec{u}$, is the velocity, ${P}$ is the pressure, and ${T}$ is the
temperature everywhere in the domain. The Reynolds number (Re), Froude number
(Fr), and Prandtl Number (Pr) are constants set for each simulation. Scaled
fluid properties like, $\rho^{\prime}$, represent the local value of the phase
scaled by the corresponding value in liquid. Therefore, $\rho^{\prime}$ is $1$
in liquid phase, and $\rho_{v}/\rho_{l}$ for vapor phase. The effect of
surface tension is modeled using Weber number (We) and incorporated by a sharp
pressure jump, ${S}^{\Gamma}_{P}$, at the liquid-vapor interface, $\Gamma$.
The effects of evaporation and saturation conditions on velocity and
temperature, ${\vec{S}^{\Gamma}_{u}}$, and ${S}^{\Gamma}_{T}$, are modeled
using a ghost fluid method [12]. For a more detailed discussion of non-
dimensional parameters and values, we refer the reader to Appendix D.

The continuity equation is given by,
$\nabla\cdot\vec{u}=-\dot{m}\nabla(\rho^{\prime})^{-1}\cdot\vec{n}$, where the
mass transfer $\dot{m}$ is computed using local temperature gradients in
liquid and vapor phase,
$\dot{m}=\text{St}(\text{Re}\>\text{Pr})^{-1}\big{[}\nabla
T_{l}\cdot{\vec{n}^{\Gamma}}-k^{\prime}\nabla
T_{v}\cdot{\vec{n}^{\Gamma}}\big{]}$ where, ${\vec{n}^{\Gamma}}$ is the
surface normal vector to the liquid-vapor interface. The Stefan number St, is
another constant defined for the simulation and depends on the the temperature
scaling given by, $\Delta T=T_{wall}-T_{bulk}$, and latent heat of
evaporation, $h_{lv}$. Simulation data is scaled to dimensional values using
the characteristic length $l_{0}$, velocity $u_{0}$, and temperature
$({T-T_{bulk}})/{\Delta T}$ scale. Temporal integration is implemented using a
fractional step predictor-corrector formulation to enforce incompressible flow
constraints. The solver has been extensively validated and demonstrates an
overall second-order accuracy in space [12, 13].

In thermal science, _heat flux_ measured as the integral of the temperature
gradient across the heater surface ($\overline{q}=\partial T/\partial y$)
serves as a vital indicator of boiling efficiency. It reflects the
contribution from multiple sub-processes such as conduction, convection,
microlayer evaporation, and bubble induced turbulence. Identifying and
managing each sub-process’s impact to enhance $\overline{q}$ is an open
challenge [16, 24]. _Critical heat flux_ (CHF) signifies peak heat flux before
a sharp drop in efficiency occurs due to the formation of a vapor barrier (see
Figure 5b). It is arguably the most important design and safety parameter for
any heat-flux controlled boiling application [31]. Accurate heat flux modeling
and prediction of boiling crisis are paramount for the reliability of heat
transfer systems [47, 71, 53].

The simulations in this study are implemented within the Flash-X framework
[15, 12], and a dedicated environment is provided for running new simulations
333 https://github.com/Lab-Notebooks/Outflow-Forcing-BubbleML. The repository
contains example configuration files for various multiphase simulations,
including those used in this dataset. To ensure reproduciblity, a lab notework
has been designed that organizes each study using configuration files for data
curation. The lab notebook and Flash-X source code are open-source to allow
for community development and contribution, enabling creation of new datasets
beyond the scope of this paper. The simulation archives store HDF5 output
files and bash scripts that document software environment and repository tags
for reproducibility. The lab notebook also provides an option to package
Flash-X simulations as standalone Docker/Singularity containers which can be
deployed on cloud and supercomputing platforms without the need for installing
third-party software dependencies. The latter is ongoing work towards software
sustainability [11].

###  3.3 Dataset Overview

The study encompasses two types of boiling namely, pool boiling and flow
boiling. Pool boiling represents fluid confined in a tank above a heater,
resembling scenarios like cooling nuclear waste. The BCs for pool boiling
include walls on the left and right, an outlet at the top, and a heater at the
bottom. In contrast, flow Boiling models water flowing through a channel with
a heater, simulating liquid cooling of data center GPUs. There is an inlet BC
modeling flow into the system and an outlet. The fluid used for the
simulations is FC-72 (perfluorohexane), an electrically insulating and stable
fluorocarbon-based fluid commonly used for cooling applications in electronics
operating at low temperatures (ranging from $50^{\circ}$C to $100^{\circ}$C).
To explore various phenomena, different parameters such as heater temperature,
liquid temperature, inlet velocity, and gravity scale are adjusted in each
simulation. A summary of the dataset is presented in Table 1. Appendix E
provides detailed illustrations of the boundary conditions and descriptions of
each simulation for reference.

Table 1: Summary of BubbleML datasets and their parameters. $\Delta t$ is the temporal resolution in non-dimensional time ($\Delta t=1=0.008$ seconds). For rationale behind the parameter choices, refer to appendix A.3. PB: pool boiling. FB: flow boiling. Dim | Type - Physics | Sims | Domain | Resolution | Timesteps | Size
---|---|---|---|---|---|---
|  |  | ($mm^{d}$) | Spatial | $\Delta t$ |  | (GB)
2D | PB - Single Bubble | 1 | $4.2\times 6.3$ | $192\times 288$ | 0.5 | 500 | 0.5
2D | PB - Saturated | 13 | $11.2\times 11.2$ | $512\times 512$ | 1 | 200 | 24.2
2D | PB - Subcooled | 10 | $8.4\times 8.4$ | $384\times 384$ | 1 | 200 | 10.3
2D | PB - Gravity | 9 | $11.2\times 11.2$ | $512\times 512$ | 1 | 200 | 16.5
2D | FB - Inlet Velocity | 7 | $29.4\times 3.5$ | $1344\times 160$ | 1 | 200 | 10.7
2D | FB - Gravity | 6 | $35\times 3.5$ | $1600\times 160$ | 1 | 200 | 10.9
2D | PB - Subcooled${}_{\text{0.1}}$ | 15 | $8.4\times 8.4$ | $384\times 384$ | 0.1 | 2000 | 155.1
2D | PB - Gravity${}_{\text{0.1}}$ | 9 | $11.2\times 11.2$ | $512\times 512$ | 0.1 | 2000 | 163.8
2D | FB - Gravity${}_{\text{0.1}}$ | 6 | $35\times 3.5$ | $1600\times 160$ | 0.1 | 2000 | 108.6
3D | PB - Earth Gravity | 1 | $8.75^{3}$ | $400^{3}$ | 1 | 57 | 122.2
3D | PB - ISS Gravity | 1 | $8.75^{3}$ | $400^{3}$ | 1 | 29 | 62.6
3D | FB - Earth Gravity | 1 | $35\times 3.5^{2}$ | $1600\times 160^{2}$ | 1 | 55 | 93.9

BubbleML stores simulation output in HDF5 files. Each HDF5 file corresponds to
the state of a simulation at a specific instant in time and can be directly
loaded into popular tensor types (e.g., PyTorch tensors or NumPy arrays) using
BoxKit. BoxKit is a custom Python API designed for efficient management and
scalability of block-structured simulation datasets [18, 10]. It leverages
multiprocessing and cache optimization techniques to improve read/write
efficiency of data between disk and memory. Figure 2 provides an example of a
boiling dataset and the corresponding workflow for enabling downstream tasks
like scientific machine learning and optical flow. By operating on simulation
data in manageable chunks that fit into memory, BoxKit significantly improves
computational performance, particularly when handling large quantities of
datasets.

Figure 2: Dataset Curation and Workflow. Flash-X multiphase simulations are
executed and converted into unblocked HDF5 formats using the BoxKit library.
The resulting dataset is publicly available 1, enabling downstream tasks like
scientific machine learning and optical flow.

Each simulation within the BubbleML dataset tracks the velocities in the x and
y directions, temperature, and a signed distance function (SDF), $\phi$, which
represents the distance from the bubble interface. The SDF can be used to get
a mask of the bubble interfaces or determine if a point is in the liquid or
vapor phase. These variables are stored in HDF5 datasets. For instance, the
temperature is stored in a tensor with a shape $t\times x\times y\times z$ ,
which allows indexing with $xyz$-spatial coordinates or time. For 2D
simulation datasets, the shape becomes $t\times x\times y$. The HDF5 files
also include any constants or runtime parameters provided to the simulation.
Some of these parameters, such as thermal conductivity or Reynolds number, are
constants used in the PDEs that govern the system. The inclusion of these
variables and parameters in the dataset enables comprehensive analysis and
modeling of the boiling phenomena.

BubbleML follows the FAIR data principles [65] as outlined in appendix A.2. To
ensure accuracy of scientific simulations, it is also essential to validate
against experimental observations due to inherent approximations in numerical
solvers and simplified models of real-world phenomena. Appendix A.4 provides a
comprehensive validation of the BubbleML dataset.

##  4 Benchmarks of BubbleML: Optical Flow and SciML

###  4.1 Optical Flow

Generation of Optical Flow Dataset. Optical flow computes the velocity field
of an image based on the relative movement of objects between consecutive
frames. This method holds significant implications for downstream tasks, such
as extracting side-view boiling statistics and applying SciML to real world
experimental data. Although many datasets capturing spatiotemporal dynamical
systems can be repurposed to create optical flow datasets, the inherent non-
rigidity of bubbles introduces unique physical phenomena that are not
prevalent in other datasets. For instance, consider the scenario where a
bubble detaches from a heater surface: the bottom region of the bubble
exhibits significantly higher velocity compared to the top region, resulting
in a velocity gradient that forces the bubble into a spherical shape. At
hotter heater temperatures, deformation and detachment process might occur
more frequently, leading to different flow patterns and bubble behaviors
illustrated in Figure 1.

We create an optical flow dataset from BubbleML to capture multiphase
phenomena. Using the liquid-vapor phase distinction information from
simulations, we distinguish between liquid and vapor phases. This enables the
generation of image sequences wherein bubble trajectories are tracked across
consecutive timesteps. Importantly, our data extraction is limited to bubble
velocities per timestep, excluding fluid velocities. This focuses the learning
task on capturing discernible objects (bubbles). The bubble velocities in non-
dimensional units are converted to pixels per frame units (see appendix B.1),
before being written to the widely used Middlebury [2] flow format, producing
a sequence of images and flow files that resemble the Sintel dataset [7]. To
facilitate the training and validation of optical flow models, PyTorch
dataloaders are provided for the generated dataset 1. This allows for easy
integration and fine-tuning of existing optical flow models using the BubbleML
dataset.

Learning Bubble Dynamics. We evaluate and fine-tune two state-of-the-art
optical flow models, RAFT [62] and GMFlow [68], using the BubbleML optical
flow dataset (B). We consider three different pre-trained models for each
method: the first model is trained exclusively on FlyingChairs (C), the second
trained on FlyingChairs and FlyingThings3D (C+T), and the third model which is
fine-tuned for the Sintel Benchmark (C+T+S). To assess the performance of the
trained models, we measure the end-point error and Table 2 summarizes the
results for one dataset. Refer to Appendix B for results on the other
datasets.

Table 2: Results of pre-trained and fine-tuned RAFT and GMFlow models on optical flow data (B) generated from PB-Saturated dataset. A 80:20 split results in 2000 training and 500 validation image pairs. The pre-trained models include C trained on FlyingChairs dataset, C+T trained further on FlyingThings3D, and C+T+S further fine-tuned for the Sintel test benchmark. Model+B represents models fine-tuned on the BubbleML optical flow dataset. Model | Method | Chairs (Val) | Things (Val) | Sintel (Train) | KITTI (Train) | Boiling (Test)
---|---|---|---|---|---|---
Clean | Clean | Final | F1-EPE | F1-all
C | RAFT | 0.82 | 9.03 | 2.19 | 4.49 | 9.83 | 37.57 | 4.20
GMFlow | 0.92 | 10.23 | 3.22 | 4.43 | 17.82 | 56.14 | 4.73
C+B | RAFT | 0.91 | 11.22 | 2.55 | 5.16 | 13.7 | 44.44 | 2.33
GMFlow | 1.31 | 11.99 | 3.78 | 5.12 | 21.91 | 63.04 | 2.36
C+T | RAFT | 1.15 | 4.39 | 1.40 | 2.71 | 5.02 | 17.46 | 4.72
GMFlow | 1.26 | 3.48 | 1.50 | 2.96 | 11.60 | 35.62 | 7.98
C+T+B | RAFT | 1.28 | 7.69 | 1.69 | 2.95 | 9.96 | 23.61 | 2.38
GMFlow | 1.39 | 3.88 | 1.61 | 2.91 | 14.49 | 43.09 | 2.51
C+T+S | RAFT | 1.21 | 4.69 | 0.77 | 1.22 | 1.54 | 5.64 | 8.39
GMFlow | 1.53 | 4.09 | 0.95 | 1.28 | 3.04 | 13.61 | 14.65
C+T+S+B | RAFT | 1.37 | 6.59 | 0.89 | 1.60 | 1.83 | 6.44 | 2.34
GMFlow | 1.65 | 4.49 | 1.07 | 1.45 | 4.06 | 18.99 | 2.56

Initially, the pre-trained models exhibit subpar performance on the BubbleML
data. To address this, each model is fine-tuned for 3-4 epochs with a low
learning rate of $10^{-6}$. After fine-tuning, we observe a significant
improvement in predictions for the test data (see Figures 7 and 8 in Appendix
B). While all fine-tuned models tend to converge to similar levels of accuracy
for pool boiling datasets, fine-tuning the pre-trained FlyingChairs models (C)
on BubbleML (B) dataset gives the best results. This could be attributed to
the similar nature of the datasets consisting of 2D objects in motion. In the
case of flow boiling, the best results are achieved by fine-tuning models
initially trained for the Sintel benchmark (C+T+S). Flow boiling images have
an extremely high aspect ratio (8:1) which is similar to the Sintel (3:1) and
the KITTI (4:1) datasets. Note that although training the models on the
boiling dataset for more epochs improves performance on our specific task, it
adversely affects the models’ generalization capabilities, leading to
increased errors on the other datasets.

Open problems. Error analysis (B.3) highlights the shortcomings of state-of-
the-art optical flow models in accurately capturing the turbulent dynamics of
bubbles. Although fine-tuning improves the overall performance, the high
errors at the bubble boundaries remain an ongoing challenge. This underscores
the need for novel optical flow models that incorporate physical insights to
accurately capture the complex and chaotic behavior of boiling. BubbleML
bridges the gap for physics-informed optical flow datasets.

###  4.2 Scientific Machine Learning

SciML Prelimaries. Our SciML baseline experiments use _neural PDE solvers_ to
learn temperature and flow dynamics. We focus on two classes of neural PDE
solvers: (a) Image-to-image models, widely used in computer vision tasks, such
as image segmentation [49]. These are not always suitable as PDE solvers,
since they can only be used on a fixed resolution, but are still competitive
in many baselines [19, 32]. (b) Neural operators are neural networks that
learn a mapping between infinite dimensional function spaces. As they map
functions to functions, neural operators are discretization invariant and can
be used on a higher resolution than they were trained [26]. The seminal neural
operator is the Fourier Neural Operator (FNO) [30]. Further details can be
found in Appendix C.

For both classes of models, we employ the auto-regressive formulation of a
forward propagator, denoted as $\mathcal{F}$. For timesteps
$\\{t_{1},\dots,t_{max}\\}$ discretized such that $t_{k+1}-t_{k}=\Delta t$,
the forward propagator $\mathcal{F}$ maps the solution function $u$ at $k$
consecutive time steps $\\{t_{m-k},\dots,t_{m-1}\\}$ to the solution at time
$t_{m}$. For brevity, we use
$u([t_{m-k},t_{m-1}])=\\{u(t_{m-k}),\dots,u(t_{m-1})\\}$. The operator
$\mathcal{F}$ can be approximated using a neural network
$\mathcal{F}_{\theta}$ parameterized by $\theta$. This network is trained
using a dataset of $N$ ground truth solutions
$D=\\{u^{(n)}([0,t_{max}]):n=1\dots N\\}$. By applying a standard gradient
descent algorithm, we find parameters $\hat{\theta}$ minimizing some loss
function of the predictions
$\mathcal{F}_{\hat{\theta}}\\{u^{(n)}([t_{m-k},t_{m-1}])\\}$ and the ground
truth solutions $u^{(n)}(t_{m})$. Thus, given solutions for $k$ initial
timesteps of an unseen function $u$, we can obtain an approximation
$\mathcal{F}_{\hat{\theta}}\\{u([0,t_{k-1}])\\}\approx u(t_{k}=t_{k-1}+\Delta
t)$. Using this approximation for $t_{k}$, we can step forward to get
$\mathcal{F}_{\hat{\theta}}\\{u([t_{1},t_{k}])\\}\approx u(t_{k+1})$. This
process is called _rollout_ and is repeated until reaching $t_{max}$. While in
principle, rollout can be done for arbitrary time, the quality of
approximation worsens with each step [32, 36]. We implement several strategies
that attempt to mitigate this deterioration [6, 63]. However, achieving long
and stable rollout is still an open problem.

Baseline Implementations. We implement several baseline image-to-image
models—including UNet${}_{\text{bench}}$ and UNet${}_{\text{mod}}$—and neural
operators—including FNO, UNO, F-FNO, and G-FNO. Detailed descriptions and
comparisons of the models are included in Appendix C.1.

Training Strategies. Detailed descriptions for each of the training stragies
we used are listed in Appendix C.2. We implement teacher-forcing training
[66], temporal bundling, and the pushforward trick [6]. Models trained with
the pushforward trick are prefixed with “P-”. A discussion of hyperparameter
settings can be found in Appendix C.3.

Metrics. We draw inspiration from PDEBench and adopt a large set of metrics
that include the Root Mean Squared Error (RMSE), Max Squared Error, Relative
Error, Boundary RMSE (BRMSE), and low/mid/high Fourier errors [61]. These
metrics provide a comprehensive view of the physical dynamics, which may be
missed when only using a “global” loss metric. For instance, when predicting
temperature, we find that the max error can often be very high due to the
presence of sharp transitions between hot vapor and cool liquid. Even a one-
pixel misalignment in the model’s prediction can cause the reported
temperature to be the opposite extreme. Metrics which report a global average
(i.e., RMSE) could mask these errors because they get damped by the average.
We incorporate an additional _physics_ metric: the RMSE along bubble
interfaces (IRMSE). The accuracy along both the domain and immersed boundaries
is very important. Boundary conditions determine if the solution to a PDE
exists and is unique. In the case of the multiphysics BubbleML dataset,
accurate modeling of the system requires satisfying the conditions at the
liquid-vapor interfaces accurately.

Learning Temperature Dynamics. One application of SciML using the BubbleML
dataset is to learn the dynamics of temperature propagation within a system.
In this context, the system’s velocities serve as a sourcing function,
influencing the temperature distribution. Notably, UNet-based models perform
best across all datasets (see Figure 3c and d). For a full listing of error
metrics for each model and dataset pairing, refer to Appendix C.4. UNet models
may have some advantage in predicting the interfaces and boundaries (IRMSE and
BRMSE), because they naturally act as edge-detectors. The temperature also
propagates smoothly, so it is likely unnecessary to use global filters, like
the FNO variants. In contrast, FNO models rely on fast Fourier transforms and
weight multiplication in the Fourier space, which, while capable of handling
global and local information simultaneously, might not be as effective at
capturing local, non-smooth features. Several recent studies report similar
observations about auto-regressive UNet and FNO variants [32, 19, 36].

Figure 3: Temperature and Heat Flux Prediction. (a) Cross-validated heatflux
$q/q_{max}$ estimates for subcooled and saturated boiling. (b), (c), and (d)
show results for the fully trained forward propagator. In (b), accuracy
degradation is minimal, with spikes occurring during timesteps of violent
turbulence caused by rapid bubble detachment from the heater surface. (c) and
(d) compare frames from the Flash-X simulation and predictions by the forward
propagator for subcooled boiling.

The trained model can be a valuable tool to get fast estimates of heat flux,
discussed in section 3.2. Heat flux is influenced by steep temperature
gradients and dynamic temporal changes which presents a challenging problem.
To further validate our models, we perform cross-validation to predict the
heatflux trends observed in Figure 5. For each heat flux prediction, we
holdout a simulation and train a forward propagator on the remaining
simulations within the dataset. Even with partial training—50 epochs for
subcooled boiling models and 100 epochs for saturated boiling models—we
achieve compelling results. The heatflux predictions by
UNet${}_{\text{bench}}$ remarkably track the expected trend, as seen in Figure
3a.

Learning Fluid Dynamics. As an additional benchmark, we use the BubbleML
dataset to train models to approximate both velocity _and_ temperature
dynamics. This is a challenging problem. Results are shown in Appendix C.5.
These follow similar training settings to the temperature-only models.
Strikingly, however, we get nearly the opposite results to predicting only
temperature: the UNet${}_{\text{bench}}$ model struggles when predicting both
velocity and temperature fields jointly, while the UNet${}_{\text{mod}}$ and
the FNO variants perform comparatively better. All of the models have
difficulty capturing the trails of condensation that form in the temperature
field. The vapor trails form, but dissipate more quickly than expected. An
example rollout of UNet${}_{\text{mod}}$, trained using the pushforward trick,
is shown in Figure 4. We see that the flow closely aligns with the ground-
truth simulation.

Figure 4: Velocity and Temperature Rollout. The left figure shows the first 80
timesteps of P-UNet${}_{\text{mod}}$’s rollout, where color indicates velocity
and streamlines illustrate direction of flow. Both the flow magnitude and
direction align exceptionally well with the ground truth. On the right, (a)
and (b) show the rollout errors for temperature and velocity predictions. The
prefix “P-” denotes that the model is trained with the pushforward trick [6].
Notably, UNet${}_{\text{mod}}$ starts with slightly better initial accuracy,
but it degrades more quickly than the model trained with the pushforward
trick. P-UNet${}_{\text{mod}}$ behaves more stably during rollout.

Open Problems. We reiterate several open problems in SciML that BubbleML
offers an avenue to explore. The first is the creation of a new class of
_models that can learn multiple interrelated physics_. We find that while UNet
architectures work well at predicting temperature and FNO variants work well
at predicting velocity, neither excel at joint prediction of temperature and
velocity. The CNN-based UNet architectures outperform FNO and its variants
when predicting temperature, potentially due to CNN’s capacity to naturally
act as edge-detectors, and thus handle non-smooth interfaces more easily. On
the other hand, FNO variants perform quite well at predicting velocities, but
still struggle with temperature estimation, especially in capturing
condensation trails. This is related to the second problem: _developing neural
operators that can handle non-smooth and irregularly shaped interfaces_. FNO
variants seem to encounter difficulties in modeling temperature fields, which
have sharp jumps along bubble interfaces where the temperature transitions
from cool liquid to hot vapor. Conversely, the velocity field appears
relatively smooth, and thus may be composed of lower frequencies better
captured by FNO variants. However, these models still miss sharp and sudden
changes in velocity along bubble interfaces that are important for accurately
modeling long-range dynamics. The third problem is improving _stability during
long rollouts_. This is explored within the context of other datasets [32,
36], but it is particularly relevant for BubbleML. For instance, in subcooled
boiling, after bubbles depart from the surface, they undergo condensation and
generate vortices that gradually dissipate as they move upstream. To model
these extended temporal processes accurately, autoregressive models must be
stable across long rollouts. However, we observe that models experience
instability, leading them to slowly diverge from the ground truth. The
BubbleML dataset presents an opportunity to study these challenges in SciML.

##  5 Conclusions and Limitations

This paper presents the BubbleML dataset, which fills a critical gap in ML
research for multiphase multiphysics systems. By employing physics-driven
simulations, the dataset offers precise ground truth information for a variety
of boiling scenarios, encompassing a wide range of parameters and providing a
comprehensive and diverse collection of data. BubbleML is validated against
experimental observations and trends, establishing its reliability and
relevance in multiphysics phase change research. The two BubbleML benchmarks
demonstrate applications in improving the accuracy of optical flow estimation
and SciML modeling encountered in multiphase systems. Importantly, BubbleML
extends its impact beyond its immediate applications. It resonates with
broader challenges in SciML, serving as a foundational platform to study
several open problems.

#### Limitations.

Combining datasets might pose challenges due to their varying sizes. Because
the resolution scales proportionally with the domain size, the constant
relative spacing between grid cells allows the UNet model to be effectively
trained on the merged boiling dataset. However, this approach does not extend
to FNO, requiring domain decomposition methods [64] or downscaling strategies
[48] to accommodate variable domain sizes. Note that the dataset is also
exclusively composed of simulations due to the unavailability of experimental
data with velocity, pressure, and temperature fields. Future work will involve
collaboration with experimentalists to augment the dataset.

##  6 Acknowledgments and Disclosure of Funding

This work was partially supported by the National Science Foundation (NSF)
under the award number 1750549, the Office of Naval Research (ONR) under grant
number N00014-22-1-2063 (supervised by program manager Dr. Mark Spector), the
Exascale Computing Project (17-SC-20-SC), a collaborative effort of the US
Department of Energy Office of Science and the National Nuclear Security
Administration, and Laboratory Directed Research and Development Program at
Argonne National Lab, Office of Science, of the U.S. Department of Energy
under Contract No. DE-AC02-06CH11357. We gratefully acknowledge the GPU
computing resources provided on HPC3, a high-performance computing cluster
operated by the Research Cyberinfrastructure Center at the University of
California, Irvine.

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##  Appendix A Additional BubbleML details

###  A.1 Dataset URLs and Links

Dataset: Links to download all the BubbleML datasets in Table 1 are available
on the GitHub homepage. The dataset homepage is hosted at
https://hpcforge.github.io/BubbleML/. All future versions of the dataset with
the new links will be uploaded here.

Code: Code for training and evaluation of all the benchmark models are
available within the same GitHub repository.

Model Weights: Weights for all the benchmark models are available in the model
zoo. All the relevant benchmark results can be accessed on the same page.

DOI: The BubbleML dataset has a DOI from Zenodo:
https://doi.org/10.5281/zenodo.8039786.

Documentation: We provide detailed descriptions for the fields tracked in the
simulation data:
https://github.com/HPCForge/BubbleML/blob/main/bubbleml_data/DOCS.md. The
documentation discusses the layout of the data, important metadata, and some
potential pitfalls.

Tutorials: We provide example Jupyter notebook to enable reproducibility of
the benchmarks and findings in this study:
https://github.com/HPCForge/BubbleML/tree/main/examples. These tutorials cover
accessing simulation data, dataset schema, querying simulation parameters, and
training a Fourier Neural Operator—discussed in Section C.1—using PyTorch.

###  A.2 Maintenance and Long Term Preservation

The authors of BubbleML are committed to maintaining and preserving this
dataset. It is likely that the authors will make extensions as we use BubbleML
for our own research. Ongoing maintenance also encompass tracking and
resolving issues identified by the broader community after release. User
feedback will be closely monitored via the GitHub issue tracker. All data is
hosted on AWS, which guarantees reliable and stable storage. Depending on
usage we may migrate to archival storage for long-term preservation.

Findable: All data is stored in an Amazon AWS S3 instance. All present and
future data will share a global and persistent DOI
https://doi.org/10.5281/zenodo.8039786.

Accessible: All data and descriptive metadata can be downloaded from the
public links listed on the GitHub homepage. For added convenience, we provide
a bash script for users to download the entire dataset at once.

Interoperable: All BubbleML data is provided in the form of standard HDF5
files that can be read using many common libraries, such as h5py for Python.
Relevant metadata is stored with each simulation.

Reusable: BubbleML is released under the Creative Commons Attribution 4.0
International License.

###  A.3 Justification of simulation parameters

We discuss the rationale guiding the selection of simulation parameters in
Table 1, addressing both the quantity of simulations and their
resolutions/time-steps.

Number of simulations: The studies performed in BubbleML encompass diverse
two-phase boiling phenomena. The number of simulations is determined based on
the distribution of the variable being studied, keeping other factors
constant. For example, in the case of saturated boiling, we choose a wall
temperature range from $60^{\circ}C$ to $120^{\circ}C$, with uniform intervals
of $5^{\circ}C$, resulting in 13 simulations. This range captures the boiling
transition from the bubbly regime to the slug regime as the heat flux
approaches criticality. However when studying the effects of gravity, the
scaling factor, $\text{Fr}^{-2}$ (Fr is the Froude Number), is chosen from the
range $10^{-4}$ to 1 using a logarithmic scale resulting in 9 simulations.
This scale is essential to cover the vastly different gravity conditions
spanning from the Earth’s surface to the International Space Station.

Domain size: Exploring phenomena across varying scales and geometries is
common practice. Such variations in sizes enable the exploration of a broad
spectrum of heat transfer dynamics, bubble dynamics, and phase change
behaviors. The domain and heater sizes are chosen to replicate typical ranges
in experiments [46, 45] while also taking into account the computational costs
of simulations.

Spatial resolution: The domain resolution depends on the grid size of an
individual 2D block. A block with spatial dimensions of $0.5\times 0.5$ (in
non-dimensionalized units) is discretized into a grid of size $16\times 16$.
This resolution is determined based on grid sensitivity studies conducted for
a single bubble case to ensure high-fidelity simulations [12]. This results in
the spatial resolution sizes in Table 1.

Temporal resolution and timesteps: The temporal discretization in the majority
of BubbleML datasets is set at 1 non-dimensional unit, equivalent to 0.008
seconds for FC-72. Additionally, we include several datasets with a finer
discretization of 0.1 non-dimensional time. In both cases, we intentionally
chose a discretization that is much larger than what the CFL-condition
mandates for the Flash-X solver. A significant advantage of neural PDE solvers
is their ability to maintain reasonable approximations while taking much
larger timesteps than traditional numerical simulations. However, this choice
introduces a trade-off between dataset size and ease of use. The datasets with
a 1 time unit discretization are potentially more accessible, but might
present challenges in achieving accurate results. Conversely, the datasets
with a 0.1 time unit discretization are less accessible and may require
distributed training, yet they are likely to achieve more accurate results due
to both more training data (i.e., more timesteps) and presence of fine-grained
physics.

###  A.4 Dataset Validation

#### Saturated and subcooled boiling.

We first validate two distinct boiling phenomena. Saturated boiling refers to
the state of a liquid when it reaches its boiling point, known as the
saturation temperature, $T_{sat}$. At this temperature, the liquid is in
equilibrium with its vapor phase, and bubbles start forming at the heated
surface. The liquid is on the verge of vaporization, and any further increase
in temperature can lead to the formation of vapor bubbles. This bubble
formation is called nucleate boiling; a far more effective way to transfer
heat than natural convection on its own. On the other hand, subcooled boiling
is a complex process with evaporation and condensation occurring
simultaneously. The heat flux imposed on the wall produces a thermal layer
around it in which bubbles may nucleate and grow. However, condensation occurs
as a bubble migrates into the bulk liquid region with temperature under
saturation point, $T_{bulk}<T_{sat}$. Experimental findings [4] indicate that
the heat flux increases linearly with the heater temperature, $T_{wall}$ until
reaching CHF, marking the transition from nucleate pool boiling to film
boiling.

Figure 5: Bubble dynamics for saturated and subcooled pool boiling at
different wall temperatures. (a) ONB during saturated boiling marked by a
bubbly flow regime. (b) CHF behavior, indicated by the transition to slug
flow. (c) Normalized heat flux, $\overline{q}/\overline{q}_{max}$, plotted
against temperature difference between heater and liquid saturation
temperature ($T_{sat}$ = $58^{\circ}C$) aligns with the boiling curve for
FC-72 in [4].

Figure 5 presents the bubble dynamics in two different regimes of saturated
pool boiling: onset of nucleate boiling (ONB) and CHF. ONB occurs at low wall
superheat and exhibits structured bubbly flow with consistent shape and size
of departing bubbles from the heater surface. In contrast, the CHF regime is
characterized by chaotic slug flow. Remarkably, the heat flux, $\overline{q}$
at the heater surface for both subcooled and saturated boiling shown in Figure
5c closely match the experimental boiling curve, specifically for Si(100)
surface [4]. Beyond CHF, the heat flux reaches a plateau, indicating a stasis
marked by presence of large pockets of vapor cover on the heater surface as
shown in Figure 5b.

Figure 6: Gravity scaling of subcooled pool boiling: Bubble dynamics (black
lines) and temperature distribution (contours) on (a) earth gravity,
$a/g=1.0$, $t_{1}<t_{2}<t_{3}$ and (b) low gravity, $a/g=0.001$. (c) Wall heat
flux vs. gravity for 2D simulations and its comparison to the scaling model in
[46].

#### Boiling at low gravity.

Boiling is the most efficient mode of heat transfer on Earth and serves as a
cooling mechanism for various thermal applications. However, in low gravity
environments such as the International Space Station (ISS), the dynamics of
bubble growth, merger, and departure, which significantly impact thermal
efficiency, are influenced by the interplay between surface tension and
gravity. Quantifying these effects is crucial for developing phase change heat
transfer systems in such environments.

Figure 6 provides an overview of simulations that were conducted to compare
against the gravity-based heat flux model proposed in [46, 45].

These simulations cover a range of gravity levels, $Fr=1-100$. The gravity
scaling, $a/g=\frac{1}{Fr^{2}}$, is used to scale values relative to the Earth
gravity ($g$). Gravity separates pool boiling into two distinct regimes:
buoyancy dominated boiling (BDB) and surface tension dominated boiling (SDB).
The transitional acceleration, $a_{trans}$, which depends on the size of the
heater, serves as the boundary between these two regimes.

In BDB, bubbles periodically detach from the heater surface when buoyancy
takes over surface tension. Figure 6 illustrates the temporal evolution of
temperature and the liquid-vapor interface during bubble departure events for
$a/g=1$. After departing from the surface, the bubbles undergo condensation
due to subcooling, and generate vortices that gradually dissipate as they move
upstream toward the outflow. This dissipation occurs as the vapor completely
condenses into liquid. Decreasing gravitational acceleration results in larger
departing bubbles and reduces the wall heat flux, $\overline{q}$. The scaling
of $\overline{q}$ with respect to gravity,
$\overline{q}_{a}/\overline{q}_{g}$, follows the slope $m_{BDB}$. In SDB, the
dynamics are dominated by the presence of a central bubble that remains on the
heater surface and acts as a vapor sink for smaller satellite bubbles, as
depicted in Figure 6 for $a/g=0.001$. The figure also captures the transient
behavior of the central bubble which fluctuates in size due to the balance
between evaporation and condensation leading to different type of vortical
structures. The heat flux drops sharply by a value $K_{jump}$, which depends
on the size of the central bubble [13], and the slope, $m_{SDB}=0$. The
gravity scaling of heat flux computed from simulations accurately matches the
expected trend from the model [46, 45], providing validation for the
simulation results.

##  Appendix B Optical Flow

###  B.1 Dataset

The BubbleML velocity data is in non-dimensional units (refer to appendix D).
The non-dimensional velocities are first converted to real world units by
multiplying it with the characteristic velocity of the simulation. The
velocities are then multiplied by a scaling factor (number of pixels per unit
length in the image) and then divided by the frames per second value. This
converts the velocities to pixels per frame units compatible with state-of-
the-art optical flow models.

###  B.2 Benchmark Models

We consider two state-of-the-art optical flow models, RAFT [62] and GMFlow
[68].

  1. 1.

RAFT, Recurrent All-Pairs Field Transforms (RAFT), processes per-pixel
features and forms multi-scale 4D correlation volumes between all pixel pairs.
The flow field is iteratively updated through a recurrent unit that performs
lookups on the correlation volumes. RAFT begins with a feature encoder that
extracts a feature vector for each pixel. This is followed by a correlation
layer that produces a 4D correlation volume for all pairs of pixels.
Subsequently, this data is pooled to create lower resolution volumes. Finally,
a GRU-based update operator utilizes the data from the correlation volumes to
iteratively refine the flow field, starting its calculations from zero.

  2. 2.

GMFlow reformulates optical flow as a global matching problem, which
identifies the correspondences by directly comparing feature similarities. It
extracts dense features from a convolutional backbone network and uses a
transformer for feature enhancement, a correlation and softmax layer for
global feature matching, and a self-attention layer for flow propagation. It
also has a refinement step to exploit higher resolution features reusing the
GMFlow framework for residual flow estimation.

###  B.3 Learning Bubble Dynamics Results

Analysis. For pool boiling test images, the models trained on FlyingChairs and
fine-tuned on Boiling images (C+B) achieve the best performance, as seen from
Tables 2, 3, and 4. However, for flow boiling test images, a different trend
emerges. As observed in Tables 5 and 6, the lowest errors are achieved by
fine-tuning the Sintel benchmark models (C+T+S+B).

Figures 7 and 8 show a visual comparison between the outputs of the best fine-
tuned and pre-trained models and the corresponding ground truth optical flows,
for RAFT and GMFlow respectively. In many instances (highlighted in red), the
pre-trained models fail to capture the direction of movement of bubbles which
are visibly corrected by fine-tuning. However, as seen in the ground truth
optical flows, the bubble boundaries generally have different values than the
interior regions. This gradient is responsible for the dynamic changes in
bubble shapes between consecutive frames. The end point errors at these
bubble-liquid interfaces remain significantly large even after fine-tuning.
The bubble-liquid interface is the most important region where a highly
accurate optical flow prediction is necessary to enable downstream tasks such
as scientific machine learning on experimental boiling data. This suggests
that there is a need to incorporate physics-informed learning rules and
architectures into optical flow models to improve the interface end point
errors in optical flow.

Table 3: Results on optical flow data (B) generated from Pool Boiling- Subcooled simulations. It consists of around 1100 training image pairs and around 300 test image pairs. It is to be noted that heater temperatures lower than $90^{\circ}C$ are not used due to the lack of trackable bubbles in many frames.  Model | Method | Chairs(Val) | Things(Val) | Sintel(Train) | KITTI(Train) | Boiling(Test)
---|---|---|---|---|---|---
Clean | Clean | Final | F1-EPE | F1-all
C | RAFT | 0.82 | 9.03 | 2.19 | 4.49 | 9.83 | 37.57 | 2.41
GMFlow | 0.92 | 10.23 | 3.22 | 4.43 | 17.82 | 56.14 | 1.92
C+B | RAFT | 0.89 | 11.64 | 2.58 | 5.17 | 14.01 | 47.53 | 0.63
GMFlow | 1.34 | 12.23 | 3.93 | 5.27 | 22.83 | 64.41 | 0.63
C+T | RAFT | 1.15 | 4.39 | 1.40 | 2.71 | 5.02 | 17.46 | 1.91
GMFlow | 1.26 | 3.48 | 1.50 | 2.96 | 11.60 | 35.62 | 4.76
C+T+B | RAFT | 1.24 | 6.45 | 1.54 | 2.82 | 7.35 | 20.27 | 0.70
GMFlow | 1.46 | 4.04 | 1.66 | 3.11 | 14.50 | 44.35 | 0.67
C+T+S | RAFT | 1.21 | 4.69 | 0.77 | 1.22 | 1.54 | 5.64 | 6.07
GMFlow | 1.53 | 4.09 | 0.95 | 1.28 | 3.04 | 13.61 | 9.21
C+T+S+B | RAFT | 1.35 | 6.49 | 0.87 | 1.48 | 1.79 | 6.31 | 0.64
GMFlow | 1.66 | 4.43 | 1.04 | 1.42 | 3.99 | 18.83 | 0.65
Table 4: Results on optical flow data (B) generated from Pool Boiling- Gravity simulations. It consists of around 1400 training image pairs and around 360 test image pairs.  Model | Method | Chairs(Val) | Things(Val) | Sintel(Train) | KITTI(Train) | Boiling(Test)
---|---|---|---|---|---|---
Clean | Clean | Final | F1-EPE | F1-all
C | RAFT | 0.82 | 9.03 | 2.19 | 4.49 | 9.83 | 37.57 | 3.30
GMFlow | 0.92 | 10.23 | 3.22 | 4.43 | 17.82 | 56.14 | 2.40
C+B | RAFT | 0.95 | 10.98 | 2.78 | 5.76 | 15.52 | 49.88 | 0.90
GMFlow | 1.42 | 12.57 | 4.06 | 5.41 | 23.61 | 66.38 | 0.90
C+T | RAFT | 1.15 | 4.39 | 1.40 | 2.71 | 5.02 | 17.46 | 2.75
GMFlow | 1.26 | 3.48 | 1.50 | 2.96 | 11.60 | 35.62 | 3.60
C+T+B | RAFT | 1.31 | 6.65 | 1.64 | 2.93 | 8.39 | 21.36 | 0.96
GMFlow | 1.41 | 3.95 | 1.64 | 2.97 | 44.07 | 14.41 | 0.97
C+T+S | RAFT | 1.21 | 4.69 | 0.77 | 1.22 | 1.54 | 5.64 | 4.74
GMFlow | 1.53 | 4.09 | 0.95 | 1.28 | 3.04 | 13.61 | 5.49
C+T+S+B | RAFT | 1.35 | 6.44 | 0.90 | 1.49 | 1.81 | 6.27 | 0.92
GMFlow | 1.63 | 4.43 | 1.03 | 1.40 | 4.29 | 20.93 | 0.92
Table 5: Results on optical flow data (B) generated from Flow Boiling- Inlet Velocity simulations. It consists of around 1200 training image pairs and around 300 test image pairs. Model | Method | Chairs(Val) | Things(Val) | Sintel(Train) | KITTI(Train) | Boiling(Test)
---|---|---|---|---|---|---
Clean | Clean | Final | F1-EPE | F1-all
C | RAFT | 0.82 | 9.03 | 2.19 | 4.49 | 9.83 | 37.57 | 16.01
GMFlow | 0.92 | 10.23 | 3.22 | 4.43 | 17.82 | 56.14 | 21.44
C+B | RAFT | 1.21 | 14.99 | 3.62 | 6.78 | 22.07 | 60.64 | 10.13
GMFlow | 1.50 | 13.96 | 4.52 | 5.89 | 23.45 | 65.96 | 7.01
C+T | RAFT | 1.15 | 4.39 | 1.40 | 2.71 | 5.02 | 17.46 | 25.14
GMFlow | 1.26 | 3.48 | 1.50 | 2.96 | 11.60 | 35.62 | 19.39
C+T+B | RAFT | 1.62 | 9.09 | 2.19 | 3.77 | 13.81 | 32.13 | 9.19
GMFlow | 1.45 | 4.15 | 1.78 | 3.05 | 15.74 | 48.34 | 7.24
C+T+S | RAFT | 1.21 | 4.69 | 0.77 | 1.22 | 1.54 | 5.64 | 21.23
GMFlow | 1.53 | 4.09 | 0.95 | 1.28 | 3.04 | 13.61 | 47.88
C+T+S+B | RAFT | 2.18 | 9.32 | 1.54 | 2.90 | 2.57 | 10.93 | 9.68
GMFlow | 1.72 | 4.80 | 1.18 | 1.65 | 4.69 | 24.56 | 6.88
Table 6: Results on optical flow data (B) generated from Flow Boiling- Gravity simulations. It consists of around 1000 training image pairs and around 250 test image pairs. Model | Method | Chairs(Val) | Things(Val) | Sintel(Train) | KITTI(Train) | Boiling(Test)
---|---|---|---|---|---|---
Clean | Clean | Final | F1-EPE | F1-all
C | RAFT | 0.82 | 9.03 | 2.19 | 4.49 | 9.83 | 37.57 | 20.42
GMFlow | 0.92 | 10.23 | 3.22 | 4.43 | 17.82 | 56.14 | 14.05
C+B | RAFT | 1.14 | 12.65 | 3.30 | 6.13 | 17.94 | 53.76 | 4.45
GMFlow | 1.30 | 13.02 | 4.05 | 5.41 | 23.17 | 65.33 | 4.24
C+T | RAFT | 1.15 | 4.39 | 1.40 | 2.71 | 5.02 | 17.46 | 21.04
GMFlow | 1.26 | 3.48 | 1.50 | 2.96 | 11.60 | 35.62 | 13.37
C+T+B | RAFT | 1.41 | 6.85 | 1.97 | 3.89 | 10.96 | 27.82 | 4.24
GMFlow | 1.37 | 3.89 | 1.71 | 3.05 | 14.29 | 45.03 | 3.94
C+T+S | RAFT | 1.21 | 4.69 | 0.77 | 1.22 | 1.54 | 5.64 | 21.04
GMFlow | 1.53 | 4.09 | 0.95 | 1.28 | 3.04 | 13.61 | 22.84
C+T+S+B | RAFT | 1.59 | 7.29 | 1.13 | 2.14 | 2.22 | 9.34 | 4.12
GMFlow | 1.62 | 4.42 | 1.04 | 1.43 | 3.86 | 17.76 | 3.93
Figure 7: Comparison of the best performing pretrained RAFT models with
finetuned RAFT models on Pool Boiling (rows 1 and 2) and Flow Boiling (row 3)
images. In each row, (a) is the ground truth optical flow, (b) the optical
flow predictions from the pretrained model, and (c) the optical flow
predictions from the finetuned model. (d) shows the normalized end-point-error
at every pixel for the pretrained model output. (e) shows the normalized end-
point-error at every pixel for the finetuned model output Figure 8: Comparison
of the best performing pretrained GMFlow models with finetuned GMFlow models
on Pool Boiling (rows 1 and 2) and Flow Boiling (row 3) images. In each row,
(a) is the ground truth optical flow, (b) the optical flow predictions from
the pretrained model, and (c) the optical flow predictions from the finetuned
model. (d) shows the normalized end-point-error at every pixel for the
pretrained model output. (e) shows the normalized end-point-error at every
pixel for the finetuned model output Figure 9: Flow field color coding used in
the above images. The displacement of every pixel is the vector from the
center of the square to this pixel.

Compute Resources. All models are fine-tuned using a single NVIDIA V100 GPU.
The implementations use the recommended versions of PyTorch and other
libraries used in the official repositories of GMFlow and RAFT.

##  Appendix C Scientific Machine Learning

###  C.1 Benchmark Models

We focus on two classes of neural PDE solvers: Image-to-image models and
neural operators. The Image-to-image models are variants of UNet, which are
commonly used in computer vision tasks, such as image segmentation. These
models are not always suitable as neural PDE solvers, where training data
generated from numerical simulations may require using different resolutions.
However, modern versions of UNet are still competitive in many benchmarks [19,
32].

  1. 1.

UNet${}_{\text{bench}}$ is a simple variant of the original UNet architecture
from 2015, which was modified in PDEBench [61] to use batch normalization. We
also replace the tanh activation with GELU [61, 49, 22].

  2. 2.

UNet${}_{\text{mod}}$ is a modern variant of UNet that gets impressive as a
PDE solver. It has been modified to use wide residual connections and group
normalization [19, 67, 70].

Neural operators have found great success as PDE solvers [26]. Typically,
given an initial condition $u_{0}$, a neural operator is a function
$\mathcal{M}:[0,t_{max}]\times\mathcal{A}\rightarrow\mathcal{A}$, where
$\mathcal{A}$ is an infinite dimensional function space, that satisfies
$\mathcal{M}(t,u_{0})=u_{t}$ [30, 26]. I.e., they map some initial condition
to any (discretized) time up to $t_{max}$. Training a model in this form
demands many valid initial conditions and solved simulations to form a large
training set. However, generating such a dataset proves infeasible in the case
of BubbleML due to the complexity and computational cost of the simulations.
To overcome this obstacle, we adopt the autoregressive approach, even for
neural operators, a practice that is also observed in [32, 36, 6].

  1. 1.

FNO is the Fourier Neural Operator [30, 26]. FNO learns weights in Fourier
space to act as a resolution-invariant global convolution. This seminal work
has been extended by many other neural operators. One notable downside is the
memory requirements: each of the Fourier domain weight matrices consume
$\mathcal{O}(H^{2}M^{D})$, where $H$ is the hidden size, $M$ is the number of
Fourier modes used after truncating high frequencies, and $D$ is the problem
dimension. In the case of BubbleML, $D$ is 2 or 3.

  2. 2.

UNO is a U-shaped variant of FNO [43]. This mimicks the skips connections of a
standard UNet, but replaces the convolutional layers with Fourier blocks. We
use an 8 layer architecture that mimicks the vanilla UNet. The encoder reduces
the input size and Fourier modes, but increases the number of channels. The
decoder increases the input size and Fourier modes, while decreasing the
number of channels. Since the number of Fourier modes decreases, much of the
network only accounts for very low frequencies.

  3. 3.

F-FNO is the Factorized Fourier Neural Operator [63]. As the name suggests,
F-FNO factorizes the Fourier transform over problem dimensions. This reduces
the number of parameters per Fourier space weight matrix to
$\mathcal{O}(H^{2}MD)$. This enables using more modes, scaling to deeper
models, or learning higher dimensional problems. Further, F-FNO suggests
changing the structure of the residual connections, so that they are applied
after the non-linearity.

  4. 4.

G-FNO is the Group-Equivariant Fourier Neural Operator [21], which extends
group convolutions to the frequency domain. The proposed G-Fourier layers are
equivariant to rotation, reflection, and translation. They keep the resolution
invariance of neural operators. While this model has many advantages, it also
has substantial memory requirements that we found limited its suitability for
high-resolution data. The Fourier space weight matrix uses
$\mathcal{O}(H^{2}M^{D}S)$, where $S$ is the number of elements in the
“stabilizer” of the group under consideration [21]. We also note that the
BubbleML simulations are _not_ rotation or vertical reflection equivariant due
to the effects of temperature and gravity. However, the simulations are
horizontal reflection and translation equivariant. In spite of this partial
incompatibility, we think G-FNO is interesting to experiment with. It is
fairly common that models that bake in some form equivariance work well, even
if the assumption of equivariance is not totally valid.

###  C.2 Training Strategies

We implement several popular training strategies designed to make neural PDE
solvers robust to instability during rollout. One source of this instability
is _distribution shift_ , which occurs when the model takes its prior outputs
as input. The errors in the model’s predictions accumulate, and the model is
likely to diverge from the ground-truth during long rollouts.

Teacher-forcing is the default training strategy [66]. The input to the model
is always the ground-truth of the previous timesteps. I.e., to predict the
solution $u(t)$, the model input is the ground-truth solutions $u([t-k,t-1])$.

Gaussian noise is applied on iterations that the pushforward trick is unused.
Several papers claim that adding noise is a simple method to help the model
become more robust to distribution shift during longer rollouts [63, 42, 50].
Similar to F-FNO, when used, we add noise with zero mean and standard
deviation of $0.01$. As Gaussian noise is spread uniformly across frequencies,
it may also help the model account for high frequencies [32].

Pushforward trick. Algorithm 1 shows sample pseudo-code for the pushforward
trick [6]. The model is trained by performing multiple inferences for
successive timesteps. Similar to using Gaussian noise, this trick helps the
model be robust to distribution shift during rollouts. The advantage is that
the errors introduced during the pushforward steps better match the
distribution of errors that accumulate during rollouts. Thus, we are
essentially training the model to correct for its own distribution shift
during rollout. We use a ramp-up when applying the pushforward trick: the
percentage of training iterations completed is used as the likelyhood that we
perform pushforward steps. So, initially there is a $0\%$ chance of pushing
forward. After $50\%$ of training iterations, there is a $50\%$ chance of
applying the pushforward trick. We found this to be necessary and that
applying the pushforward trick on every iteration would fail.

Algorithm 1 Pushforward Trick.

1:input is the model’s input; $N$ is the number of pushforward steps

2:for $n\in\\{0,\dots,N\\}$ do

3: $\text{input}\leftarrow\text{model}(\text{input})$ $\triangleright$ Step
though $N$ timesteps

4:end for

5:$\text{input}\leftarrow\text{model}(\text{input})$ $\triangleright$ Track
gradients for only the final forward pass.

Temporal bundling is used so that the model outputs the solution for multiple
future timesteps with one inference [6]. By having the model output the
solution for multiple future timesteps, we can reduce the number of times the
model is used. Since some error is introduced by each inference during
rollout, temporal bundling should reduce the rate that error is accumulated.
This is again beneficial for maintaining accuracy and stability during long
rollouts. With a temporal bundle of size $k$, the forward propagator outputs a
vector $\mathcal{F}(u([t_{n-k},t_{n-1}]))=u([t_{n},t_{n+k-1}])$.

Algorithm 2 Main Training Loop, with Pushforward Trick.

1:data, the set of simulations; $N$, the number of pushforward steps

2:sample $t$ from $\\{t_{k},t_{max}\\}$

3:$\text{input}\leftarrow\text{data}([t-k\Delta t,t-\Delta t])$
$\triangleright$ Sample $k$ timesteps to use as input

4:if $N=0$ then

5: $\text{input}\leftarrow\text{input}+\mathcal{N}(0,0.01)$ $\triangleright$
Add Gaussian noise when _not_ “pushing forward.”

6:end if

7:output = PushForwardTrick(input, $N$)

8:$\text{target}\leftarrow\text{data}([t+Nk,t+(N+1)k-1])$ $\triangleright$
Grab target for result of $N$ pushforward steps

9:$\text{loss}(\text{output},\text{target})$

###  C.3 Hyperparameter Settings

We mostly use generic hyperparameter settings that we found work well across
all models. Each model has its own parameters (hidden channels, Fourier modes,
etc) tuned based on these settings. The settings for the temperature and flow
experiments can be found in Tables 7 and 8. All models are trained using AdamW
[33]. For each temperature model, we use 3% of the total iterations for
learning rate warmup to reach the base learning rate. For the remaining
iterations, we apply a learning rate decay. For UNet${}_{\text{mod}}$, we
reduce the base learning rate to 5e-4 in order to avoid instability we
experienced during some training runs. We also perform gradient clipping to
clip the gradient norm to be at most one. For data augmentation, we perform
horizontal reflections only for pool boiling.

The Fourier models that are resolution invariant are trained at half
resolution and evaluated at full resolution. We found that training at the
full resolution gave similar results, but took much longer or required
distributed training. The Fourier models all take the $xy$-coordinate grid as
input [21]. So, the input is $(x,y,T([t-k,t-1]),v([t-k,t-1]))$, where $T$ is
the temperature map and $v$ is the velocity map. Each Fourier model uses the
64 lowest frequency modes, except UNO, which uses the 32 lowest frequencies in
the bottom of the “U” where the resolution is too small to use 64 modes. As
the domain is non-periodic, we do Fourier continuation by padding the domain
with zeros [30]. For G-FNO, due to memory constants, we are forced to use
fewer Fourier modes (32) and downsample the domain to quarter of its original
size during training. The G-FNO models are trained on an NVIDIA A100 with 80
GB of memory.

Table 7: Generic hyperparameter settings for the temperature experiments. Hyperparameter | Value
---|---
Number of Epochs | 250
Batch Size | 8
Optimizer | AdamW
Weight Decay | 0.01
Base LR | 1e-3 (or 5e-4)
LR Warmup | Linear, 3%
LR Scheduler | Step
Step Factor | 0.5
Step Patience | 75 Epochs
History Size | 5
Future Size | 5
Table 8: Generic hyperparameter settings for the flow experiments. Hyperparameter | Value
---|---
Number of Epochs | 25
Batch Size | 16
Optimizer | AdamW
Weight Decay | 1e-4
Base LR | 5e-4
LR Warmup | Linear, 3%
LR Scheduler | Cosine Annealing
Min LR | 1e-6
History Size | 5
Future Size | 5

###  C.4 Learning Temperature Dynamics Results

We show the full listing of results for predicting temperature dynamics in
Tables 9, 10, 11, 12, and 13. As a general trend, we see that
UNet${}_{\text{bench}}$ consistently performs the best. In some metrics,
UNet${}_{\text{mod}}$ closely matches or is better than
UNet${}_{\text{bench}}$ but it tends to have higher errors at bubble
interfaces and domain boundaries. The FNO variants perform well, but slightly
worse. The good performance of UNet architecture might stem from the nature of
convolutions to function as effective edge detectors. Thus, the UNet variants
can handle sharp interfaces between liquid and vapor. Since the Fourier-layers
implicitly assume periodicity, they tend to act as “smoothers”. Qualitatively,
they appear to soften the sharp interfaces, resulting in errors along the
region surrounding the bubbles. This likely contributes to the consistent
higher maximum, interface, and boundaries errors observed for the FNO
variants. There is existing work that makes similar observations [19, 36].
It’s important to highlight G-FNO’s particularly poor performance. This is
likely due to two reasons: (a) the substantial memory requirements prevented
us from using the same resolution and Fourier modes as the other models. (b)
BubbleML violates some of the assumptions that G-FNO makes. Specifically, the
simulations are not invariant to vertical reflections due to the effects of
temperature and gravity. For instance, if the heater was on the top of the
domain, we would not see a reflection of the phenomena that we do when it is
at the bottom.

Table 9: Temperature Prediction: Pool Boiling Subcooled. | UNet${}_{\text{bench}}$ | UNet${}_{\text{mod}}$ | FNO | UNO | FFNO | GFNO
---|---|---|---|---|---|---
Rel. Err. | 0.036 | 0.051 | 0.052 | 0.062 | 0.050 | 0.071
RMSE | 0.035 | 0.049 | 0.050 | 0.059 | 0.048 | 0.068
BRMSE | 0.073 | 0.146 | 0.118 | 0.145 | 0.124 | 0.209
IRMSE | 0.113 | 0.157 | 0.189 | 0.220 | 0.155 | 0.290
Max Err. | 2.1 | 1.937 | 2.32 | 2.659 | 2.21 | 2.882
F. Low | 0.281 | 0.348 | 0.373 | 0.440 | 0.472 | 0.557
F. Mid | 0.266 | 0.386 | 0.390 | 0.453 | 0.375 | 0.547
F. High | 0.040 | 0.059 | 0.055 | 0.068 | 0.057 | 0.090
Table 10: Temperature Prediction: Pool Boiling Saturated. | UNet${}_{\text{bench}}$ | UNet${}_{\text{mod}}$ | FNO | UNO | FFNO | GFNO
---|---|---|---|---|---|---
Rel. Err. | 0.035 | 0.039 | 0.052 | 0.072 | 0.053 | 0.066
RMSE | 0.035 | 0.039 | 0.052 | 0.071 | 0.052 | 0.065
BRMSE | 0.082 | 0.147 | 0.165 | 0.214 | 0.163 | 0.218
IRMSE | 0.052 | 0.083 | 0.125 | 0.116 | 0.095 | 0.152
Max Err. | 1.701 | 1.785 | 2.055 | 2.23 | 1.788 | 1.793
F. Low | 0.257 | 0.241 | 0.373 | 0.568 | 0.416 | 0.463
F. Mid | 0.268 | 0.273 | 0.377 | 0.532 | 0.399 | 0.496
F. High | 0.023 | 0.042 | 0.053 | 0.069 | 0.052 | 0.070
Table 11: Temperature Prediction: Pool Boiling Gravity. | UNet${}_{\text{bench}}$ | UNet${}_{\text{mod}}$ | FNO | UNO | FFNO | GFNO
---|---|---|---|---|---|---
Rel. Err. | 0.042 | 0.051 | 0.062 | 0.081 | 0.061 | 0.124
RMSE | 0.040 | 0.049 | 0.062 | 0.077 | 0.058 | 0.118
BRMSE | 0.070 | 0.139 | 0.150 | 0.184 | 0.131 | 0.642
IRMSE | 0.108 | 0.201 | 0.266 | 0.335 | 0.239 | 0.424
Max Err. | 2.92 | 2.846 | 3.626 | 3.667 | 3.22 | 4.0
F. Low | 0.491 | 0.466 | 0.500 | 0.746 | 0.578 | 1.095
F. Mid | 0.275 | 0.326 | 0.440 | 0.548 | 0.423 | 0.760
F. High | 0.033 | 0.049 | 0.054 | 0.077 | 0.049 | 0.171
Table 12: Temperature Prediction: Flow Boiling Gravity. | UNet${}_{\text{bench}}$ | UNet${}_{\text{mod}}$ | FNO | UNO
---|---|---|---|---
Rel. Err. | 0.055 | 0.095 | 0.134 | 0.157
RMSE | 0.051 | 0.088 | 0.123 | 0.144
BRMSE | 0.080 | 0.240 | 0.267 | 0.301
IRMSE | 0.091 | 0.214 | 0.351 | 0.379
Max Err. | 2.560 | 2.800 | 3.951 | 3.990
F. Low | 0.263 | 0.388 | 0.591 | 0.602
F. Mid | 0.281 | 0.419 | 0.551 | 0.692
F. High | 0.149 | 0.276 | 0.389 | 0.449
Table 13: Temperature Prediction: Flow Boiling Inlet Velocity. All models have substantially higher error compared to the other datasets. This is likely caused by the higher velocity. Since the temporal discretization is quite coarse, each bubble will take a large “jump” between frames that is difficult to capture accurately. | UNet${}_{\text{bench}}$ | UNet${}_{\text{mod}}$ | FNO | UNO
---|---|---|---|---
Rel. Err. | 0.106 | 0.122 | 0.202 | 0.225
RMSE | 0.097 | 0.111 | 0.184 | 0.205
BRMSE | 0.271 | 0.315 | 0.486 | 0.471
IRMSE | 0.178 | 0.193 | 0.252 | 0.319
Max Err. | 2.900 | 2.862 | 3.527 | 3.992
F. Low | 0.571 | 0.584 | 1.061 | 0.981
F. Mid | 0.511 | 0.566 | 1.084 | 1.334
F. High | 0.264 | 0.315 | 0.494 | 0.523

###  C.5 Learning Flow Dynamics Results

Table 14 shows the predicted temperature, $x$ velocity, and $y$ velocity
during rollout of 200 timesteps on the Subcooled Pool Boiling study. We opted
to exclude G-FNO from this experiment due to its immense memory requirements.
We observe an interesting phenomenon: UNet${}_{\text{bench}}$ goes from being
the best model at predicting temperature, to being among the worst. This is
the simplest model, so it is understandable that its limited capaticity
prevents it from learning more complex functions. In comparison,
UNet${}_{\text{mod}}$ performs incredibly well. Surprisingly, the FNO variants
perform quite well. The best performing model is P-UNet${}_{\text{mod}}$,
which was trained using the pushforward trick. It achieves the best score in
all metrics, except, surprisingly, for the maximum error. UNO experiences some
divergence. Initially, it makes good predictions (perhaps better than the UNet
architectures), but over long rollouts, its predictions tend towards infinity.
Since UNO is the model that experiences the most striking divergence, we
compare its 1D radially averaged power spectrum with UNet${}_{\text{mod}}$ and
P-UNet${}_{\text{mod}}$ in Figure 10. We are able to reproduce findings that
show that autoregressive Fourier models experience aliasing errors, causing
them to diverge [36].

Figure 10: 1D Radially Averaged Power Spectrum of the simulation temperature and model rollout temperature. We see that initially, all of the models follow the simulations frequency fairly well, but UNO quickly diverges. This phenomena has been reported [36] as being aliasing errors. The plots are on log-scale, so curves below the simulations are more accurate than curves above the simulation. Table 14: Coupled Velocity and Temperature Prediction: Pool Boiling Subcooled0.1. P-UNet${}_{\text{mod}}$ was trained using the pushforward trick. It gets the best results during rollout. Notably, FFNO perform better than UNet${}_{\text{bench}}$ and UNet${}_{\text{mod}}$. This is a sharp contrast to the temperature predictions, where the UNet variants were far better. UNO diverged during rollout. This is likely because is mostly uses low frequencies. The error in the high frequencies accumulates during rollout, and it diverges. | Metric | UNet${}_{\text{bench}}$ | UNet${}_{\text{mod}}$ | P-UNet${}_{\text{mod}}$ | FNO | UNO | FFNO
---|---|---|---|---|---|---|---
Temp. | Rel. Err. | 0.108 | 0.074 | 0.040 | 0.066 | 0.322 | 0.067
RMSE | 0.105 | 0.072 | 0.039 | 0.064 | 0.051 | 0.065
BRMSE | 0.248 | 0.211 | 0.113 | 0.142 | 0.050 | 0.237
IRMSE | 0.178 | 0.191 | 0.129 | 0.212 | 0.077 | 0.201
Max Err. | 2.83 | 2.340 | 2.741 | 2.566 | 7.492 | 2.488
F. Low | 1.407 | 0.495 | 0.244 | 0.531 | 4.902 | 0.498
F. Mid | 0.740 | 0.583 | 0.280 | 0.518 | 1.594 | 0.507
F. High | 0.090 | 0.076 | 0.051 | 0.061 | 0.442 | 0.083
$x$ Vel. | Rel. Err. | 0.681 | 0.553 | 0.429 | 0.642 | 1.834 | 0.547
RMSE | 0.021 | 0.015 | 0.012 | 0.018 | 0.051 | 0.015
BRMSE | 0.023 | 0.023 | 0.022 | 0.023 | 0.050 | 0.024
IRMSE | 0.051 | 0.049 | 0.043 | 0.049 | 0.077 | 0.050
Max Err. | 0.294 | 0.320 | 0.353 | 0.32 | 0.574 | 0.334
F. Low | 0.329 | 0.225 | 0.142 | 0.258 | 0.677 | 0.189
F. Mid | 0.115 | 0.105 | 0.079 | 0.110 | 0.328 | 0.107
F. High | 0.012 | 0.012 | 0.011 | 0.012 | 0.092 | 0.012
$y$ Vel. | Rel. Err. | 0.563 | 0.400 | 0.311 | 0.415 | 1.170 | 0.397
RMSE | 0.021 | 0.015 | 0.012 | 0.016 | 0.044 | 0.015
BRMSE | 0.015 | 0.012 | 0.010 | 0.011 | 0.035 | 0.013
IRMSE | 0.046 | 0.046 | 0.038 | 0.045 | 0.066 | 0.047
Max Err. | 0.292 | 0.272 | 0.280 | 0.290 | 0.288 | 0.289
F. Low | 0.387 | 0.190 | 0.132 | 0.180 | 0.660 | 0.176
F. Mid | 0.128 | 0.112 | 0.085 | 0.111 | 0.240 | 0.106
F. High | 0.011 | 0.010 | 0.009 | 0.010 | 0.067 | 0.010

##  Appendix D Physical and non-dimensional values

###  D.1 Fluid properties

Table 15 provides a comprehensive list of physical properties relevant to
multiphase flow. These properties are presented with their respective symbols,
units, and descriptions. It is important to consider the properties of the
liquid and vapor phases separately in multiphase flows, as they exhibit
distinct characteristics due to the phase change phenomenon. Additionally, the
temperatures of the heater and liquid are also taken into account, as they
significantly influence bubble dynamics and flow behaviors. The saturation
temperature, which marks the initiation of bubble generation, is determined by
the specific fluid being studied.

Table 15: Physical properties with symbol and unit Symbol | Unit | Description
---|---|---
$\rho_{l}$ | $kg/m^{3}$ | Liquid density
$\rho_{v}$ | $kg/m^{3}$ | Vapor density
$\mu_{l}$ | $N\cdot s/m^{2}$ | Dynamic viscosity of liquid
$\mu_{v}$ | $N\cdot s/m^{2}$ | Dynamic viscosity of vapor
$C_{p,l}$ | $J/kg\cdot K$ | Specific heat capacity of liquid
$C_{p,v}$ | $J/kg\cdot K$ | Specific heat capacity of vapor
$k_{l}$ | $W/m\cdot K$ | Thermal conductivity of liquid
$k_{v}$ | $W/m\cdot K$ | Thermal conductivity of vapor
$h_{vl}$ | $J/kg$ | Latent heat
$g$ | $m/s^{2}$ | Gravitational acceleration
$\sigma$ | $N/m$ | Surface tension
$T_{wall}$ | $K$ | Heater temperature
$T_{bulk}$ | $K$ | Bulk temperature
$T_{sat}$ | $K$ | Saturation temperature

###  D.2 Conversion to Non-dimensional parameters

Table 16 provides a comprehensive list of non-dimensional variables used in
this study, including their symbols, definitions, and descriptions. For
example, the non-dimensional time is obtained by dividing the characteristic
length scale by the characteristic velocity scale. These non-dimensional
variables are necessary to solve the non-dimensionalized governing equations
such as the continuity, momentum, and energy equations. Additionally, non-
dimensional properties such as density, dynamic viscosity, specific heat
capacity, and thermal conductivity are also considered in the context of
multiphase flow. Notably, representative parameters in fluid mechanics,
thermodynamics, and heat transfer, such as the Reynolds number ($Re$), Froude
number ($Fr$), Prandtl number ($Pr$), Stefan number ($St$), Weber number
($We$), and Peclet number ($Pe$), are used to obtain simulation results. By
employing non-dimensionalization, the effects of different physical quantities
can be studied independently of their specific units, facilitating a deeper
understanding of the underlying phenomena.

Table 16: Formulae used for conversion of real world values to non-dimensional values Symbol | Definition | Description
---|---|---
$\rho^{*}$ | $\rho_{v}/\rho_{l}$ | Non-dimensional density
$\mu^{*}$ | $\mu_{v}/\mu_{l}$ | Non-dimensional viscosity
$C_{p}^{*}$ | $C_{p,v}/C_{p,l}$ | Non-dimensional specific heat capacity
$k^{*}$ | $k_{v}/k_{l}$ | Non-dimensional thermal conductivity
$l_{0}$ | $\sqrt{\sigma/g\Delta\rho}$ | Characteristic length scale
$u_{0}$ | $\sqrt{gl_{0}}$ | Characteristic velocity scale
$t_{0}$ | $l_{0}/u_{0}$ | Characteristic time scale
$T_{wall}^{*}$ | $(T_{wall}-T_{bulk})/(T_{wall}-T_{bulk})=1$ | Non-dimensional heater temperature
$T_{bulk}^{*}$ | $(T_{bulk}-T_{bulk})/(T_{wall}-T_{bulk})=0$ | Non-dimensional bulk temperature
$T_{sat}^{*}$ | $(T_{sat}-T_{bulk})/(T_{wall}-T_{bulk})$ | Non-dimensional saturated temperature
$Re$ | $\rho_{l}u_{0}l_{0}/\mu_{l}$ | Reynolds number
$Fr$ | $u_{0}/\sqrt{gl_{0}}$ | Froude number
$Pr$ | $\mu_{l}C_{p,l}/k_{l}$ | Prandtl number
$St$ | $C_{p,l}(T_{wall}-T{bulk})/h_{vl}$ | Stefan number
$We$ | $\rho_{l}u_{0}^{2}l_{0}/\sigma$ | Weber number
$Pe$ | $Re\cdot Pr$ | Peclet number

##  Appendix E Simulation Details

###  E.1 Multiphase Simulations

Numerical simulations of multiphase flows with phase changes have been studied
using various techniques to model the behavior at the liquid-vapor interface
and track its evolution over time. Two commonly used methods for handling
boundary conditions related to surface tension and evaporation are the ghost
fluid method (GFM) and the continuum surface force method (CSF). The GFM
enforces a sharp jump in pressure, velocity, and temperature across the
interface, while the CSF diffuses the forcing within the vicinity of the
interface for a smoother transition. The choice between these approaches
involves a trade-off between accuracy and stability, with GFM offering higher
accuracy but lower stability compared to CSF. Interface tracking is typically
achieved implicitly using level-set or volume of fluid (VOF) techniques.

Researchers have employed these methods to study and model various aspects of
multiphase flows with phase changes. For example, Gibou et al. [17] used a
level-set method with sharp interfacial jump conditions within the framework
of the GFM to model homogeneous two-dimensional evaporation and film boiling.
Son and Dhir [54, 55] extended this approach to perform heterogeneous pool
boiling calculations involving single and multiple bubbles. Majority of their
initial work focused on model development and verification using two-
dimensional (2D) simulations, since real world three-dimensional (3D)
calculations were expensive due to limitations of the software framework.

Efforts have also been made to perform high-fidelity 3D simulations of pool
boiling by combining different techniques. Yazadani et al. [69] conducted
critical heat flux (CHF) calculations on earth gravity using a combination of
VOF and CSF methods. Sato et al. [52, 51], on the other hand used the level-
set method in combination with CSF for their simulations. These studies
highlighted the computational cost associated with boiling simulations which
had to be mitigated by performing low resolution calculations. More recently,
Dhruv et al. [12, 13] used a combination of level-set and GFM methods for
gravity scaling analysis of pool boiling at finer resolution than previous
studies using adaptive mesh refinement (AMR) within the framework of FLASH.
These simulations were applied to study effects of gravity on boiling heat
transfer which lead to verification of experiment based heat flux models and
enabled the quantification of turbulent heat flux associated with bubble
dynamics during bubbly and slug flow [13]. The implementation of multiphase
models within FLASH has transitioned to Flash-X, which leverages state-of-art
AMR techniques and heterogeneous supercomputing architectures to significantly
improve performance of boiling calculations. The BubbleML dataset is curated
from simulations carried out with the Flash-X framework.

An important note is that simulations still heavily rely on experimental
observations to determine input conditions such as nucleation site
distribution, bubble nucleation frequency, and solid-liquid-vapor contact
angle dynamics [12]. As a result, simulations serve as an effective tool to
understand and quantify trends in boiling regimes, rather than attempting to
replicate experiments precisely. This opens up an opportunity for the
integration of data-driven ML techniques, which can leverage diverse datasets
to make informed predictions for boiling phenomena.

###  E.2 Fluid parameters

The input configuration file of a simulation requires the inclusion of certain
parameters, which remain constant for a specific fluid. For our simulations
using FC-72 (Perfluorohexane, $C_{6}F_{14}$), the non-dimensional values for
this fluid are provided in Table 17. It is important to note that parameters
corresponding to any specific real-world fluid must be converted to these non-
dimensional values before being input into the simulation configuration files.
This conversion allows for consistent and standardized representation of fluid
properties in the simulation, enabling accurate and meaningful results to be
obtained.

Table 17: Non-dimensional constants for FC-72, the fluid used in BubbleML Parameter | Variable Name | Non-dimensional Value
---|---|---
Inverse Reynolds Number, $\frac{1}{Re}$ | ins_invReynolds | 0.0042
Non-dimensional density, $\rho^{*}$ | mph_rhoGas | 0.0083
Non-dimensional viscosity, $\mu^{*}$ | mph_muGas | 1.0
Non-dimensional thermal conductivity, $k^{*}$ | mph_thcoGas | 0.25
Non-dimensional specific heat capacity, $C_{p}^{*}$ | mph_CpGas | 0.83
Inverse Weber Number, $\frac{1}{We}$ | mph_invWeber | 1.0
Prandtl Number, $Pr$ | ht_Prandtl | 8.4

##  Appendix F BubbleML Datasheet

In this section, we provide details about the BubbleML dataset published with
a permanent doi 10.5281/zenodo.8039786.

###  F.1 Motivation

  1. 1.

For what purpose was the dataset created? Was there a specific task in mind?
Was there a specific gap that needed to be filled? Please provide a
description.

The dataset was created to enable machine learning driven research on phase
change phenomena by providing a comprehensive, open source collection of high
fidelity Boiling Simulations.

  2. 2.

Who created this dataset (e.g. which team, research group) and on behalf of
which entity (e.g., company, institution, organization)?

The dataset was primarily created by Sheikh Md Shakeel Hassan, Arthur Feeney
(PhD students at HPCForge, University of California Irvine) in collaboration
with Akash Dhruv(Argonne National Laboratory) and Jihoon Kim(Korea
University).

  3. 3.

Who funded the creation of the dataset? If there is an associated grant,
please provide the name of the grant or and the grant name and number.)

The work was funded by the National Science Foundation (NSF) under the award
number 1750549.

  4. 4.

Any other comments?

No.

###  F.2 Composition

  1. 1.

What do the instances that comprise the dataset represent (e.g. documents,
photos, people, countries)? Are there multiple types of instances (e.g.
movies, users, and ratings; people and interactions between them; nodes and
edges)? Please provide a description.)

Each instance of the dataset represents a study of pool boiling or flow
boiling where a particular parameter such as heater temperature or gravity is
varied over a range of values.

  2. 2.

How many instances are there in total (of each type, if appropriate)?

There are a total of 6 boiling studies done in 2D, comprising of 4 pool
boiling and 2 flow boiling studies. The dataset also contains 2 3D pool
boiling simulations. For details of each study please refer Table 1.

  3. 3.

Does the dataset contain all possible instances or is it a sample(not
necessarily random) of instances from a larger set? If the dataset is a
sample, then what is the larger set? Is the sample representative of the
larger set (e.g. geographic coverage)? If so, please describe how this
representativeness was validated/verified. If it is not representative of the
larger set, please describe why not (e.g., to cover a more diverse range of
instances, because instances were withheld or unavailable).)

It is practically impossible to cover all possible instances of Multiphase
Simulations. Our dataset encompasses a wide range of physical phenomena that
can be seen in boiling fluids. Nevertheless, we provide a framework for new
data generation3 using the same tools that we have used in this paper.

  4. 4.

What data does each instance consist of? ("Raw" data (e.g. unprocessed text or
images) or features? In either case, please provide a description.)

Each instance consists of a number of hdf5 simulation files. Each such file
contains velocity, temperature, pressure and liquid-vapor phase information in
numpy arrays of dimensions $(t,x,y)$.

  5. 5.

Is there a label or target associated with each instance? If so, please
provide a description.

Each simulation is associated with a set of physical runtime parameters. They
are included in each hdf5 simulation file and can be used for training
purposes.

  6. 6.

Is any information missing from individual instances? (If so, please provide a
description, explaining why this information is missing (e.g., because it was
unavailable). This does not include intentionally removed information, but
might include, e.g., redacted text.)

No.

  7. 7.

Are relationships between individual instances made explicit (e.g., users’
movie ratings, social network links)? (If so, please describe how these
relationships are made explicit.)

The studies are independent from each other.

  8. 8.

Are there recommended data splits (e.g., training, development/validation,
testing)? (If so, please provide a description of these splits, explaining the
rationale behind them.)

The training, validation and testing splits can be at the end-users discretion
depending on the downstream task the dataset is used for.

  9. 9.

Are there any errors, sources of noise, or redundancies in the dataset? (If
so, please provide a description.)

Small discretization errors are inevitable in numerical simulations. However,
the simulations capture the trends seen in experimental data and the errors do
not affect the correctness of the dataset.

  10. 10.

Is the dataset self-contained, or does it link to or otherwise rely on
external resources (e.g., websites, tweets, other datasets)? (If it links to
or relies on external resources, a) are there guarantees that they will exist,
and remain constant, over time; b) are there official archival versions of the
complete dataset (i.e., including the external resources as they existed at
the time the dataset was created); c) are there any restrictions (e.g.,
licenses, fees) associated with any of the external resources that might apply
to a future user? Please provide descriptions of all external resources and
any restrictions associated with them, as well as links or other access
points, as appropriate.)

Yes.

  11. 11.

Does the dataset contain data that might be considered confidential (e.g.,
data that is protected by legal privilege or by doctor-patient
confidentiality, data that includes the content of individuals’ non-public
communications)? (If so, please provide a description.)

No.

  12. 12.

Does the dataset contain data that, if viewed directly, might be offensive,
insulting, threatening, or might otherwise cause anxiety? (If so, please
describe why.)

No.

  13. 13.

Does the dataset relate to people? (If not, you may skip the remaining
questions in this section.)

No.

  14. 14.

Does the dataset identify any subpopulations (e.g., by age, gender)? (If so,
please describe how these sub-populations are identified and provide a
description of their respective distributions within the dataset.)

N/A.

  15. 15.

Is it possible to identify individuals (i.e., one or more natural persons),
either directly or indirectly (i.e., in combination with other data) from the
dataset? (If so, please describe how.)

N/A.

  16. 16.

Does the dataset contain data that might be considered sensitive in any way
(e.g., data that reveals racial or ethnic origins, sexual orientations,
religious beliefs, political opinions or union memberships, or locations;
financial or health data; biometric or genetic data; forms of government
identification, such as social security numbers; criminal history)? (If so,
please provide a description.)

N/A.

  17. 17.

Any other comments?

No.

###  F.3 Collection Process

  1. 1.

How was the data associated with each instance acquired? (Was the data
directly observable (e.g., raw text, movie ratings), reported by subjects
(e.g., survey responses), or indirectly inferred/derived from other data
(e.g., part-of-speech tags, model-based guesses for age or language)? If data
was reported by subjects or indirectly inferred/derived from other data, was
the data validated/verified? If so, please describe how.)

The multiphase simulations were performed using Flash-X[15]. Some example
configuration files for the simulations provided in the dataset can be found
here3. The simulations have been validated against real world experimental
trends previously established in literature. Please refer to section A.4 for
details.

  2. 2.

What mechanisms or procedures were used to collect the data(e.g.,hardware
apparatus or sensor, manual human curation, software program, software API)?
(How were these mechanisms or procedures validated?)

The simulations were run on the HPC3 cluster in University of California
Irvine.

  3. 3.

If the dataset is a sample from a larger set, what was the sampling strategy
(e.g., deterministic, probabilistic with specific sampling probabilities)?

Since it is impossible to capture all possible physical conditions of a
Multiphysics system, studies were selected to capture a wide range of physical
phenomena such as varying heater temperature, gravity and flow velocity.

  4. 4.

Who was involved in the data collection process (e.g., students, crowd
workers, contractors) and how were they compensated (e.g., how much were crowd
workers paid)?

Data generation was done by the authors.

  5. 5.

Over what timeframe was the data collected? (Does this timeframe match the
creation timeframe of the data associated with the instances (e.g., recent
crawl of old news articles)? If not, please describe the timeframe in which
the data associated with the instances was created.)

Simulations were created over the time period April-June 2023.

  6. 6.

Were any ethical review processes conducted (e.g., by an institutional review
board)? (If so, please provide a description of these review processes,
including the outcomes, as well as a link or other access point to any
supporting documentation.)

No.

  7. 7.

Does the dataset relate to people? (If not, you may skip the remaining
questions in this section.)

No.

  8. 8.

Did you collect the data from the individuals in question directly, or obtain
it via third parties or other sources (e.g., websites)?

N/A.

  9. 9.

Were the individuals in question notified about the data collection? (If so,
please describe (or show with screenshots or other information) how notice was
provided, and provide a link or other access point to, or otherwise reproduce,
the exact language of the notification itself.)

N/A.

  10. 10.

Did the individuals in question consent to the collection and use of their
data? (If so, please describe (or show with screenshots or other information)
how consent was requested and provided, and provide a link or other access
point to, or otherwise reproduce, the exact language to which the individuals
consented.)

N/A.

  11. 11.

If consent was obtained, were the consenting individuals provided with a
mechanism to revoke their consent in the future or for certain uses? (If so,
please provide a description, as well as a link or other access point to the
mechanism (if appropriate).)

N/A.

  12. 12.

Has an analysis of the potential impact of the dataset and its use on data
subjects (e.g., a data protection impact analysis) been conducted? (If so,
please provide a description of this analysis, including the outcomes, as well
as a link or other access point to any supporting documentation.)

N/A.

  13. 13.

Any other comments?

None.

###  F.4 Preprocessing/cleaning/labeling

  1. 1.

Was any preprocessing/cleaning/labeling of the data done(e.g.,discretization
or bucketing, tokenization, part-of-speech tagging, SIFT feature extraction,
removal of instances, processing of missing values)? (If so, please provide a
description. If not, you may skip the remainder of the questions in this
section.)

The raw data obtained from the simulations are in a block structured format
for efficient memory management in 3D simulations. For 2D simulations, the
block structure can be abandoned as a single unblocked simulation is only 2-3
GB in size. This makes data access easier. We provide the 2D simulations in an
unblocked format as they are easier to read and load into existing PyTorch and
Tensorflow workflows.

  2. 2.

Was the "raw" data saved in addition to the preprocessed/cleaned/labeled data
(e.g., to support unanticipated future uses)? (If so, please provide a link or
other access point to the "raw" data.)

The raw simulation data is saved on the cluster used to run the simulations.
However we do not anticipate any future uses for the raw data as the data
contained in the unblocked simulations is essentially the same, but the raw
unblocked data can be made available upon request

  3. 3.

Is the software used to preprocess/clean/label the instances available? (If
so, please provide a link or other access point.)

The preprocessing code is available in the github repository1.

  4. 4.

Any other comments?

None.

###  F.5 Uses

  1. 1.

Has the dataset been used for any tasks already? (If so, please provide a
description.)

The 3D simulations have been used in previous papers which have been
appropriately cited in this paper and a few papers are a work in progress.

  2. 2.

Is there a repository that links to any or all papers or systems that use the
dataset? (If so, please provide a link or other access point.)

Not currently.

  3. 3.

What (other) tasks could the dataset be used for?

The dataset is primarily intended for the scientific machine learning
community. The dataset could probably be used for other studies such as bubble
segmentation. We invite the community to use the dataset in innovative ways.

  4. 4.

Is there anything about the composition of the dataset or the way it was
collected and preprocessed/cleaned/labeled that might impact future uses? (For
example, is there anything that a future user might need to know to avoid uses
that could result in unfair treatment of individuals or groups (e.g.,
stereotyping, quality of service issues) or other undesirable harms (e.g.,
financial harms, legal risks) If so, please provide a description. Is there
anything a future user could do to mitigate these undesirable harms?)

No.

  5. 5.

Are there tasks for which the dataset should not be used? (If so, please
provide a description.)

No.

  6. 6.

Any other comments?

None.

###  F.6 Distribution

  1. 1.

Will the dataset be distributed to third parties outside of the entity (e.g.,
company, institution, organization) on behalf of which the dataset was
created? (If so, please provide a description.)

Yes, the dataset is freely and publicly available and accessible.

  2. 2.

How will the dataset will be distributed (e.g., tarball on website, API,
GitHub)? (Does the dataset have a digital object identifier (DOI)?)

The dataset is free for download by everyone. Links are available in the
github repository1. The doi of the dataset is
https://doi.org/10.5281/zenodo.8039786.

  3. 3.

When will the dataset be distributed?

The dataset is distributed as of June 2023 in its first version.

  4. 4.

Will the dataset be distributed under a copyright or other intellectual
property (IP) license, and/or under applicable terms of use (ToU)? (If so,
please describe this license and/or ToU, and provide a link or other access
point to, or otherwise reproduce, any relevant licensing terms or ToU, as well
as any fees associated with these restrictions.)

The dataset is licensed under CC BY license.

  5. 5.

Have any third parties imposed IP-based or other restrictions on the data
associated with the instances? (If so, please describe these restrictions, and
provide a link or other access point to, or otherwise reproduce, any relevant
licensing terms, as well as any fees associated with these restrictions.)

No

  6. 6.

Do any export controls or other regulatory restrictions apply to the dataset
or to individual instances? (If so, please describe these restrictions, and
provide a link or other access point to, or otherwise reproduce, any
supporting documentation.)

No

  7. 7.

Any other comments?

None.

###  F.7 Maintenance

  1. 1.

Who is supporting/hosting/maintaining the dataset?

The dataset is being maintained by the High Performance Computing
laboratory(HPCForge) of the EECS department in University of California
Irvine.

  2. 2.

How can the owner/curator/manager of the dataset be contacted (e.g., email
address)?

The manager of the dataset can be reached at<EMAIL_ADDRESS>
  3. 3.

Is there an erratum? (If so, please provide a link or other access point.)

Currently, there is no erratum. If errors are encountered, the dataset will be
updated with a fresh version. They will all be provided in the same github
repository1.

  4. 4.

Will the dataset be updated (e.g., to correct labeling errors, add new
instances, delete instances’)? (If so, please describe how often, by whom, and
how updates will be communicated to users (e.g., mailing list, GitHub)?)

Same as above.

  5. 5.

If the dataset relates to people, are there applicable limits on the retention
of the data associated with the instances (e.g., were individuals in question
told that their data would be retained for a fixed period of time and then
deleted)? (If so, please describe these limits and explain how they will be
enforced.)

N/A.

  6. 6.

Will older versions of the dataset continue to be supported/hosted/maintained?
(If so, please describe how. If not, please describe how its obsolescence will
be communicated to users.)

Versioning of the dataset will be maintained in the github repository and the
homepage.

  7. 7.

If others want to extend/augment/build on/contribute to the dataset, is there
a mechanism for them to do so? (If so, please provide a description. Will
these contributions be validated/verified? If so, please describe how. If not,
why not? Is there a process for communicating/distributing these contributions
to other users? If so, please provide a description.)

Flash-X is a publicly available software and simulations can be generated by
anyone with the available compute resources. Examples are provided 3.

  8. 8.

Any other comments?

None.

###  F.8 Reproducibility of the baseline score

The training details and models are provided in the BubbleML github
repository1 for reproducibility of reported results.

###  F.9 Reading and using the dataset

The datasets are presented as compressed tarballs, one for each study. Upon
decompressing, a number of HDF5 files are obtained which can be read by any
standard library(e.g., h5py for Python). For code examples in python, please
refer to the github repository1. Scripts are provided for loading the data
into scientific machine learning models and creating Optical Flow datasets.

###  F.10 Data Format

Each HDF5 file has the following keys: "dfun", "int-runtime-params",
"pressure", "real-runtime-params", "temperature", "velx", "vely", "x" and "y".
"dfun", "pressure", "temperature", "velx", "vely", "x" and "y" are numpy
arrays of size (Temporal Dimension, X-Spatial Dimension, Y-Spatial Dimension).
These keys contain the liquid-vapor phase information, pressure, temperature,
velocities in x and y direction and the x and y coordinates in non-dimensional
units respectively. The keys "int-runtime-params" and "real-runtime-params"
contain various physical constants(Reynolds Number, Prandtl Number etc.) that
were used in the respective simulations.