# Continual Learning of Range-Dependent Transmission Loss for Underwater
Acoustic using Conditional Convolutional Neural Net

Indu Kant Deo<EMAIL_ADDRESS>Akash Venkateshwaran
<EMAIL_ADDRESS>Rajeev K. Jaiman<EMAIL_ADDRESS>Department of
Mechanical Engineering, The University of British Columbia, Vancouver, BC V5T
1Z4, Canada

###### Abstract

There is a significant need for precise and reliable forecasting of the far-
field noise emanating from shipping vessels. Conventional full-order models
based on the Navier-Stokes equations are unsuitable, and sophisticated model
reduction methods may be ineffective for accurately predicting far-field noise
in environments with seamounts and significant variations in bathymetry.
Recent advances in reduced-order models, particularly those based on
convolutional and recurrent neural networks, offer a faster and more accurate
alternative. These models use convolutional neural networks to reduce data
dimensions effectively. However, current deep-learning models face challenges
in predicting wave propagation over long periods and for remote locations,
often relying on auto-regressive prediction and lacking far-field bathymetry
information. This research aims to improve the accuracy of deep-learning
models for predicting underwater radiated noise in far-field scenarios. We
propose a novel range-conditional convolutional neural network that
incorporates ocean bathymetry data into the input. By integrating this
architecture into a continual learning framework, we aim to generalize the
model for varying bathymetry worldwide. To demonstrate the effectiveness of
our approach, we analyze our model on several test cases and a benchmark
scenario involving far-field prediction over Dickin’s seamount in the
Northeast Pacific. Our proposed architecture effectively captures transmission
loss over a range-dependent, varying bathymetry profile. This architecture can
be integrated into an adaptive management system for underwater radiated
noise, providing real-time end-to-end mapping between near-field ship noise
sources and received noise at the marine mammal’s location.

††preprint: AIP/123-QED

##  I Introduction

Marine vessel activities are a major source of underwater ocean noise, which
poses a significant threat to marine ecosystems Duarte  _et al._ (2021).
Solving this problem requires a collaborative, interdisciplinary approach
involving environmentalists and engineers. Recent advancements in naval
architecture and engineering offer promising ways to reduce noise through
innovative green vessel design and operational strategies. Intelligent Green
Marine Vehicles (IGMVs) are crucial in addressing the issue of underwater
ocean noise. These next-generation IGMVs need to be adaptable and continuously
learn from their environment to adjust their operations. For example, one of
the effective operational strategies is nearly real-time optimization of noise
emissions from marine vessels, particularly in areas frequented by marine
mammals Venkateshwaran  _et al._ (2024). Moreover, they must continually learn
about the varied bathymetry environments encountered during operations. This
real-time prediction capability empowers vessels to adjust their operations
dynamically to minimize acoustic disturbance to marine life. The illustration
in Figure 1 depicts a scenario of ship operation integrating real-time noise
prediction, displaying the significance of this capability in mitigating
environmental impact.

Figure 1: Illustration of real-time noise prediction from a marine vessel at a
distant mammal location.

Forecasting the underwater noise generated by marine vessels requires a
thorough understanding of the dynamics of underwater radiated noise (URN).
Underwater ocean noise represents a complex wave propagation phenomenon,
involving interactions between acoustic wavefronts and the ocean’s surface and
floor Jensen  _et al._ (2011). The dynamics of underwater acoustic wave
propagation are governed by hyperbolic partial differential equations (PDEs),
which stem from conservation laws LeVeque  _et al._ (2002). Consequently,
there has been increasing attention within the research community towards
developing efficient and accurate computational solutions for these underlying
PDEs. Utilizing advanced numerical discretization techniques to solve these
partial differential equations can result in significant computational
overheads, making them impractical for tasks requiring real-time prediction.
To create an adaptable marine system, it is crucial to have the capability to
predict the far-field noise spectrum generated by vessels within complex and
dynamic environments in nearly real-time. Conventional wisdom dictates that a
lower-dimensional reduced-order model, which can effectively capture the
physics of the problem and produce solutions in nearly real-time, should
replace the full-order PDE model.

Conventional approaches employ the wave equation to create simplified models
for predicting sound propagation in the ocean, aiming to enhance computational
efficiency. Examples of such methods include ray models Porter and Bucker
(1987); Porter (2019) and normal mode models Williams (1970); Chapman and Ward
(1990). These models require various environmental factors like sound speed,
seabed properties, and bathymetry as input to predict accurate sound
propagation. While physics-based acoustic solvers offer valuable insights,
they pose challenges to integrate data from different sources and with varying
levels of accuracy, such as simulation data and experimental measurements.
Consequently, it is essential to build a flexible data-driven model capable of
assimilating ocean acoustics data from various sources and environmental
conditions. Such a general-purpose model could be employed to evaluate
underwater ocean acoustic propagation for varying bathymetry and different
ocean environment conditions. Nevertheless, creating generalized physical
models for underwater ocean acoustics is a difficult task due to the complex
multi-physics phenomena involved, the wide range of physical scales, and the
uncertainties associated with numerous environmental parameters James and
Dowling (2005, 2011). This research focuses on building a data-driven physics-
based machine learning algorithm designed to forecast noise originating from
distributed sources at the expected locations of marine mammals in nearly
real-time.

Machine learning has experienced a resurgence in recent years, largely due to
the impressive accomplishments of deep learning (DL) models in diverse fields.
LeCun, Bengio, and Hinton (2015). One area experiencing a notable resurgence
is reduced-order modeling, which utilizes deep learning models to approximate
complex physical systems Deo and Jaiman (2022); Deo, Gao, and Jaiman (2023);
Gao, Deo, and Jaiman (2024). Among the numerous deep learning models
available, deep neural networks stand out as a highly effective technique for
representing and modeling physical phenomena, as exemplified by their
successful application in modeling complex systems like the Navier-Stokes
equations Yang, Yang, and Xiao (2016). We propose utilizing a data-driven
method based on deep learning to construct a model of underwater sound
propagation. The data-driven approach relies solely on the data, independent
of the method used to generate the data. As a result, this can effectively
approximate the underlying ocean acoustic environment across diverse
environmental conditions and varying ocean bathymetry, utilizing data acquired
from experiments or computational solvers. Moreover, these data-driven models
can distill a much lower-dimensional representation of the system from high-
dimensional physical data, enabling real-time predictions.

Recently, a variety of data-driven models, including DL models, have been
utilized in underwater acoustics for classifying and localizing ship-generated
sound sources based on their acoustic characteristics in both shallow and deep
ocean environments Niu, Ozanich, and Gerstoft (2017); Huang  _et al._ (2018);
Chi  _et al._ (2019); Ferguson (2021). Additionally, DL models have been
increasingly employed to investigate shallow water wave propagation Fotiadis
_et al._ (2020); Deo and Jaiman (2022). However, some of these approaches
explicitly rely on partial differential equations Borrel-Jensen, Engsig-Karup,
and Jeong (2021), making them less suitable for handling high-dimensional
physical data and complex systems where the exact governing equations and
underlying physical processes may be unknown. Of particular interest, Mallik
et al. applied a convolutional recurrent autoencoder network (CRAN) for
predicting transmission loss Mallik, Jaiman, and Jelovica (2022). CRAN Bukka
_et al._ (2021); Deo and Jaiman (2022) operates similarly to numerical
discretization methods. It encodes high-dimensional physical data into a
latent space and subsequently propagates it along the temporal dimension,
analogous to discretization methods that discretize physical space and use
time integration for solution marching. This methodology addresses the
challenges related to knowing the underlying differential equation and
mitigates the computational burden associated with numerical methods. However,
CRAN-type architecture struggles to predict far-field noise due to their
autoregressive prediction where information from the previous step is used to
predict the next step. Since these architectures lack information on the whole
ocean bottom’s geometry, they are not able to predict underwater radiated
noise at distant locations with accuracy.

To tackle these challenges, we present a novel range-dependent conditional
convolutional neural network (RC-CAN) designed to predict transmission loss
accurately based on input ocean bottom geometry. Our method aims to achieve a
single-shot prediction, thus alleviating the limitations of autoregressive
prediction methods. The RC-CAN model is a composite encoder-decoder framework,
which transforms the input ocean bathymetry mesh using an encoder network and
uses a decoder network to decode the transmission loss on the input mesh. The
RC-CAN is employed for learning the spatial distribution of far-field
transmission loss from a point source in a two-dimensional underwater ocean
environment with varying bathymetry. We consider a wide range of varying ocean
bathymetry to demonstrate the generalized learning capacity of the RC-CAN
model.

For the first time, we approach underwater radiated noise modeling as a
continual learning task, leveraging prior work in the field Wang  _et al._
(2024); Rebuffi  _et al._ (2017). This approach allows us to adjust to
changing ocean bathymetry while retaining previously learned information. We
introduce a novel replay-based training method for the RC-CAN model Rolnick
_et al._ (2019), marking the first application of a continual learning
framework in the context of dynamical systems prediction. The replay-based
training method entails storing a subset of data from past tasks or conditions
and retraining the model on this subset Tiwari  _et al._ (2022). This strategy
ensures that the model maintains memory of earlier tasks, mitigating
catastrophic forgetting, a common issue in sequential learning scenarios where
the model forgets previous information when trained on new data Kirkpatrick
_et al._ (2017); Bhatt, Ross, and Sigal (2024). By employing a replay-based
strategy, the RC-CAN model can sequentially train on new ocean bathymetry data
while retaining knowledge from prior training instances Ke  _et al._ (2021).
This successful implementation opens up avenues for future applications of
data-driven models in real-time forecasting of far-field underwater noise
transmission across diverse parameters and bathymetry. Integration of this
method into IGMV noise mitigation systems holds promise for continual
operation adaptation and reduction of the impact on marine mammals.

The rest of the paper is structured as follows. Section 2 covers mathematical
preliminaries and elaborates on our proposed RC-CAN network. Section 3
describes the replay training strategy used for our network architecture. In
Section 4, we perform a numerical analysis focusing on predicting far-field
underwater radiated noise using RC-CAN over a wide variety of varying
bathymetry. The paper concludes in Section 5, where we provide a brief
discussion of our findings and propose potential avenues for future research.

##  II  Mathematical Formulation

This section begins with a concise overview of data-driven reduced-order
modeling in the context of underwater radiated noise. Following this, we
introduce our range-dependent conditional convolutional neural network.

###  II.1 Data-driven reduced-order modeling

Underwater noise transmission is governed by the propagation of acoustic
pressure generated from sound sources. The propagation of acoustic pressure is
modeled using the wave equation for pressure perturbation given by:

$\rho\nabla\cdot\left(\frac{1}{\rho}\nabla
p\right)-\frac{1}{c^{2}}\frac{\partial^{2}p}{\partial t^{2}}=0,$ (1)

where $p$ and $\rho$ are pressure and density perturbations. If the density is
constant in space, Eq. 1 can be replaced by the standard form of the wave
equation,

$\nabla^{2}p-\frac{1}{c^{2}}\frac{\partial^{2}p}{\partial t^{2}}=0.$ (2)

Since the coefficients of the two differential operators in Eq. 2 are
independent of time, the dimension of the wave equation can be reduced using
frequency–time Fourier transform leading to Helmholtz equation in Cartesian
coordinates $\mathbf{x}=(x,y,z)$, given by:

$\nabla^{2}p+\frac{\omega^{2}}{c^{2}(\mathbf{x})}p=-\delta\left(\mathbf{x}-\mathbf{x}_{0}\right),$
(3)

where $c(\mathbf{x})$ is the sound speed and $\omega$ is the angular frequency
of the source located at $\mathbf{x}_{0}$. To establish the computational
domain, it is necessary to specify the boundary conditions, which are
expressed as:

$p(\partial\mathbf{x};\omega)=g(\partial\mathbf{x};\omega),$ (4)

where $\partial\mathbf{x}$ is the boundary of the domain and Eq. 4 represents
the boundary conditions of the PDE.

In a simplified two-dimensional representation of the ocean environment, the
acoustic pressure can be expressed in Cartesian coordinates as
$p(R,z;\omega)$. Here, $p(R,z;\omega)$ depends on the range $R$, depth $z$,
and the frequency of the emitted signal $\omega$. Many numerical techniques,
such as the finite-difference method (FDM), discretize the space $(R,z)$ to
obtain the solution. FDM solves the Helmholtz equation on the discretized grid
$(R_{N},z_{N})$ to compute the pressure $p(R_{N},z_{N})$ at the discretized
locations. These methods incorporate ocean bathymetry information into the
boundary conditions of the partial differential equations (PDEs). Figure 2
represents a rectangular finite-difference mesh for such applications.

Figure 2: A schematic of finite-difference approximation on a rectangular
mesh.

The acoustic energy or intensity $\left(I(z,R)\text{, and I}\propto
p^{2}\right)$ reduces significantly with the distance from the source and this
is referred to as transmission loss (TL). This loss can be measured in
decibels by comparing the acoustic pressure $p(R,z)$ at a receiver location
$(R,z)$ with the reference signal $p_{0}$ emitted by the source.

$\mathrm{TL}(R,z)=-10\log\frac{I(R,z)}{I_{0}}=-20\log\frac{p(R,z)}{p_{0}}.$
(5)

Hence the transmission loss is a measure of the drop in far-field sound
pressure level from various sources of URN. For this study, we want to build a
reduced-order model that predicts the far-field transmission loss on the input
mesh. We can employ the deep neural network to build such a reduced-order
model given by:

$\mathrm{TL}(R_{N},z_{N})=\mathcal{NN}(R_{N},z_{N};\theta),$ (6)

here, $(R_{N},z_{N})$ represents the input mesh containing data regarding
receiver locations and ocean bathymetry. $\theta$ denotes the trainable
parameters of the deep neural network. Therefore, the objective of this study
is to learn the operator $\mathcal{NN(.)}$ that transforms the input mesh
$(R_{N},Z_{N})$ with bathymetry information to transmission loss
$\mathrm{TL}(R_{N},z_{N})$ using a data-driven approach. The subsequent
section introduces the range-dependent conditional convolutional neural
network, which is utilized to construct this reduced-order model.

###  II.2 Range-dependent conditional convolutional neural network

To develop a reduced-order model for predicting acoustic transmission loss, we
will utilize a range-dependent conditional convolutional neural network based
on a 2D convolutional neural network architecture. First, a stack of
convolutional and fully connected neural networks transforms the input ocean
bathymetry mesh into a latent space. Subsequently, a 2D convolution decoder
network will be employed to learn a mapping from the low-dimensional manifold
to a high-dimensional space. This approach aims to capture the relationship
between ocean bathymetry and far-field transmission loss by applying
convolutional kernels to the input mesh representing ocean bathymetry, with
the decoder network learning the transmission loss on the input mesh. Figure 3
depicts the process of encoding input mesh into latent space and then decoding
the transmission loss on the input mesh.

Figure 3: An illustration of the proposed range conditional convolutional
network.

The encoder and decoder can be expressed as:

$\displaystyle\tilde{h}_{r}=\bm{\Psi}_{E}\left(\mathbf{R_{N},z_{N}};\theta_{E}\right)$
, (7a)
$\displaystyle\hat{\mathrm{TL}}(R_{N},z_{N})=\bm{\Psi}_{D}\left(\tilde{h}_{r};\theta_{D}\right)$
, (7b)

where $\hat{\mathrm{TL}}(R_{N},z_{N})$ denotes the approximation of the
transmission loss on the input mesh $(R_{N},z_{N})$. The ocean bathymetry and
domain are masked on the input mesh $(R_{N},z_{N})$.
$\bm{\Psi_{E}(.;\theta_{E})}$ stands for the encoder network that maps the
input ocean bathymetry to a low-dimensional manifold $\mathbf{\tilde{h}_{r}}$,
and $\theta_{E}$ is the parameter of the encoder network.
$\bm{\Psi_{D}(.;\theta_{D})}$ denotes the decoder network that maps the low-
dimensional data back to the high-dimensional physical state, and $\theta_{D}$
is the parameter of the decoder. By backpropagating the $L_{2}$ norm of error,
the weights of the neural network $(\theta_{E},\theta_{D})$, can be trained.

We outline key characteristics of convolutional neural networks that make them
suitable for reduced-order modeling in transmission loss prediction. Firstly,
underwater radiated noise involves the spread of a disturbance from its
source, with a pattern that propagates through the domain. The translational
invariance property of convolutional neural networks is beneficial for
modeling this propagation Bronstein  _et al._ (2017). Additionally,
convolutions are well-suited for learning ocean bathymetry features as they
can effectively capture local features using different convolutional kernels.

The 2D convolutional network employed in this study consists of two main
components: an encoder and a decoder path. The encoder path follows a standard
convolutional network architecture, comprising four padded convolutions, each
followed by batch normalization and Leaky ReLU activation. Subsequently, a max
pooling operation is applied to down-sample the data. Mathematically, the
encoder path is represented by Eq. 7a. The decoder path, represented by Eq.
7b, involves up-sampling the feature map followed by four transpose
convolutions (up-convolution), each also followed by batch normalization and
Leaky ReLU activation. The final layer uses a 2D convolution kernel with one
feature channel to align with the input data. Overall, the network comprises 8
convolutional layers.

By utilizing the encoder and decoder network, we can construct the reduced-
order model or learn the $\mathcal{NN(.)}$ operator as a composition of the
encoder and decoder, expressed as:

$\mathcal{NN(.)}=\bm{\Psi}_{D}\circ\bm{\Psi}_{E}(.).$ (8)

The training of this convolutional network involves finding the parameters
that minimize the expected reconstruction error over all training examples
given by

${\theta}_{E}^{*},{\theta}_{D}^{*}=\arg\min_{{\theta}_{{E}},{\theta}_{D}}\mathcal{L}[\mathrm{TL}(R_{N},z_{N}),\Psi_{D}(\Psi_{E}({R_{N},z_{N}};\theta_{E});\theta_{D})],$
(9)

where
$\mathcal{L}[\mathrm{TL}(R_{N},z_{N}),\Psi_{D}(\Psi_{E}({R_{N},z_{N}};\theta_{E});\theta_{D})]$
is a loss function in the $L^{2}$ norm, which minimizes the difference between
the reconstruction
$\left(\Psi_{D}(\Psi_{E}({R_{N},z_{N}};\theta_{E});\theta_{D})\right)$ and the
ground truth $\left(\mathrm{TL}(R_{N},z_{N})\right)$. The training of RC-CAN
is discussed in detail in the subsequent section.

##  III Replay-based Training Strategy of RC-CAN

In this section, we turn our attention to the details of creating the training
dataset and the associated training procedure. We curate a dataset containing
seven different ocean bathymetry profiles, which can be categorized into three
groups. The first group includes an idealized representation of seamounts
using triangles. Seamounts are underwater mountains that rise from the ocean
floor but do not reach the water’s surface. The second group comprises wedge
profiles, while the third group consists of bathymetry profiles sampled from
Dickin’s seamount location in the Northeast Pacific Ocean.

The idealized seamount profiles assume a flat bottom bathymetry at a depth of
3000 m, except for the area where the seamount rises. These profiles are
divided into three sub-categories. The first subgroup assumes a fixed base of
the seamount, which is 20 km wide and located 10 km from the source. The peak
of the seamount is located at a range of 20 km from the source, with the depth
of the peak varying from 500 m to 1600 m. To create the dataset for this
subgroup, 75 values of peak depth are randomly sampled between these heights.
The second subgroup assumes that the depth of the peak is either 800 m or 1200
m, while the base width of the seamount varies randomly from 5 km to 40 km.
For this subgroup, 35 values are randomly sampled between these points. The
third subgroup represents a generalized case where the peak depth and width
are randomly sampled. These three subgroups are illustrated in Figure 4.

Figure 4: Idealized seamount profiles. (a-c): fixed base width (20 km) and
peak depth (500 m to 1600 m). (d-f): peak depth (800 m or 1200 m) and base
width (5 km to 40 km). (g-i): randomly sampled peak depths and base widths.

The second group comprises wedge profiles, characterized by gradual changes in
ocean bathymetry depth, either increasing or decreasing. A special case within
the wedge profile includes both decreasing and increasing profiles within the
data range. Figure 5 shows the wedge bathymetry profiles. The third group
consists of bathymetry profiles sampled from the location of Dickin’s seamount
in the Northeast Pacific. Figure 6 depicts the bathymetry of Dickin’s
seamount. If the network architecture inherently possesses a strong implicit
bias to capture the physics of underwater radiated noise, then having a small
sample size of bathymetry instances from this location could be sufficient for
training RC-CAN in a realistic ocean environment. Therefore, we sampled only
10 bathymetries from the Dickins seamount location for training.

Figure 5: Wedge bathymetry profiles. (a) Depth decreasing profile, (b)
increasing depth profile, and (c) decreasing then increasing depth profile.
Figure 6: An image depicting a cross-sectional view of the Dickins seamount in
the range-depth plane.

Each bathymetry profile corresponds to 2049 receiver locations along the depth
and 1408 receiver locations along the range. The depth of the ocean domain is
fixed at 3000 m, enabling the examination of deep ocean sound propagation.
Furthermore, the range considered for this study spans 100 km in the domain.
The input for training consists of a mesh that covers the receiver locations,
with the mesh dimensions specified as $1408\times 2049$. In this mesh
representation, ocean bathymetry is denoted by a value of 1, while the ocean
itself is represented by a value of 0. This mesh configuration is visually
depicted in Figure 4.

The training dataset $\mathcal{D=\\{D_{\text{1}},...,D_{\text{7}}}\\}$
comprises three groups:
$\\{\mathcal{D_{\text{1}},D_{\text{2}},D_{\text{3}}}\\}$ blong to idealized
seamount group, $\\{\mathcal{D_{\text{4}},D_{\text{5}},D_{\text{6}}}\\}$
represent wedge profiles, and $\mathcal{D_{\text{7}}}$ depicts the Dickins
seamount data. We denote the task of predicting transmission loss for the
dataset $\mathcal{D_{\text{i}}}$ as Task i.

When the model is trained on the entire dataset $\mathcal{D}$, it does not
perform well in generalization or in predicting transmission loss for the
individual tasks. To address this, the model can be trained sequentially on
Task ${1}$, to Task ${7}$. Initially, training the model solely on Task
$1,2,\text{and }3$ yields exceptional performance in predicting transmission
loss for the idealized seamount. However, training on Task ${4}$ causes the
weights to be optimized for Task ${4}$, resulting in suboptimal performance on
Task $1,2,\text{and }3$. This phenomenon is known as catastrophic forgetting.

A continual learning system must exhibit both plasticity Ostapenko  _et al._
(2019), involving the acquisition of new knowledge, and stability, involving
the retention of old knowledge Kong  _et al._ (2022). Catastrophic forgetting
arises when stability breaks down, leading to new experiences overriding
previous ones. In the brain, it is widely believed that replaying past
experiences helps alleviate forgetting Rolnick  _et al._ (2019). To address
this issue, a replay buffer dataset $\mathcal{K}$ can be created. After
completing training on Task $i$, the model is re-trained for a fixed number of
epochs on $\mathcal{K_{\text{i}}}$. The replay buffer $\mathcal{K_{\text{i}}}$
contains an exemplar set
$\\{\mathcal{P_{\text{1}}},...,\mathcal{P_{\text{i-1}}}\\}$, where
$\mathcal{P_{\text{i}}}$ consists of representative examples from the dataset
$\mathcal{D_{\text{i}}}$. Several methods exist for selecting the exemplar
set, such as the gradient coreset replay buffer, which is a strategy for
selecting and updating the replay buffer using a carefully designed
optimization criterion Tiwari  _et al._ (2022). In this study, we randomly
sample examples Brahma and Othon (2018) from the dataset
$\mathcal{D_{\text{i}}}$ to create the exemplar set $\mathcal{P_{\text{i}}}$.
The complete training algorithm is presented in the subsequent sections.

For improved neural network training and to prevent the over-saturation of any
particular feature, the training transmission loss for each profile are scaled
appropriately as follows:

$\displaystyle\overline{\mathrm{TL}}_{\text{Train}}$
$\displaystyle=\frac{{\mathrm{TL}}_{\text{Train}}-{\mathrm{TL}}_{\text{Train}_{,\mu}}}{{Std.Dev.(\mathrm{TL}}_{\text{Train}})},$
(10)

where $\overline{\mathrm{TL}}_{\text{Train}}$ represents the transmission loss
data that has been normalized to have a mean of zero and a standard deviation
of one. This normalized training set is then utilized for training the model.

The range conditional convolution neural network makes predictions in a single
shot on a given input bathymetry mesh. The input dimensions are $1408\times
2049$. We want to determine the model parameters
$\theta^{*}=\left\\{\theta_{E}^{*},\theta_{D}^{*}\right\\}$ such that they
minimize the following expected error between the model prediction and the
ground truth data

$\displaystyle\mathcal{L}\left(\tilde{\mathrm{TL}},\mathrm{TL}\right)=\frac{1}{N_{m}}\sum_{j=1}^{N_{m}}\left(\left\|\tilde{\mathrm{TL}}_{j}-{\mathrm{TL}}_{j}\right\|_{2}^{2}\right),$
(11)

where $\tilde{\mathrm{TL}}$ and $\mathrm{TL}$ are the model prediction and the
ground truth respectively. While $N_{m}$ denotes the various ocean bathymetry
profiles.

In the present work, we employ a cosine annealing warm restart learning rate
scheduler. This scheduler utilizes the cosine function as the learning rate
annealing function and periodically resets or increases the learning rate
Loshchilov and Hutter (2016). We use the AdamW optimizer Loshchilov and Hutter
(2019), a version of the adaptive gradient algorithm Adam. Algorithm 1
provides a complete training procedure for our architecture.

Input: Dataset: $\mathcal{D}$, Parameters: $\theta$, Patience: $N_{p}$,
Epochs: $N_{\text{epochs}}$, Batch size: $N_{b}$, Replay Epochs:
$N_{\text{replay}}$

Output: $\\{\theta^{*}:=(\theta_{E}^{*},\theta_{D}^{*}\\}$)

Initialize Parameters: $\theta$ and Replay Buffer: $\mathcal{K}=\emptyset$;

$m\leftarrow 0//\text{ The counter for early stopping }$;

for  _ $\mathcal{D}_{\text{i}}\in\mathcal{D}$_ do

while  _epoch $<$ $N_{\text{epochs}}$ and $m<N_{p}$ _ do

$\left((R_{N},z_{N})_{\text{Train}},\mathrm{TL}_{\text{Train}}\right)\in\mathcal{D_{\text{i}}}$;

Randomly sample batch of input bathymetry from training data:
${(R_{N},z_{N})}_{\text{Train}}^{b}\subset{(R_{N},z_{N})}_{\text{Train}}$;

Forward pass:
${{\tilde{\mathrm{TL}}}}^{b}\leftarrow\Psi_{D}\left(\Psi_{E}\left({(R_{N},z_{N})}_{\text{Train}}^{b};\theta_{E}\right);\theta_{D}\right)$,
${\tilde{\mathrm{TL}}}^{b}\in\mathbb{R}^{N_{b}\times R_{N}\times z_{N}}$;

Calculate loss $\mathcal{L}$ via Eq. (11);

Estimate gradients $\hat{\mathbf{g}}$ using automatic differentiation;

Update parameters: $\theta\leftarrow ADAMW(\hat{\mathbf{g}})$;

if  _$\mathcal{L}$ $\leq$ ${l_{b}}$_ then

Save the model parameters;

$m\leftarrow 0$;

/* lb is best loss */

${l_{b}}$ $\leftarrow\mathcal{L}$;

else

$m\leftarrow m+1$

end if

end while

$\mathcal{P}_{i-1}\leftarrow\text{ConstructExemplarSet}\left(\mathcal{D}_{i-1}\right)$;

$\mathcal{K}_{i}\leftarrow\text{ConstructReplayBuffer}\left(\mathcal{P}_{1},...,\mathcal{P}_{i-1}\right)$;

while  _epoch $<$ $N_{\text{replay}}$_ do

$\left((R_{N},z_{N})_{\mathcal{K}_{i}},\mathrm{TL}_{\mathcal{K}_{i}}\right)\in\mathcal{K}_{i}$;

${(R_{N},z_{N})}_{\mathcal{K}_{i}}^{b}\subset{(R_{N},z_{N})}_{\mathcal{K}_{i}}$;

Forward pass:
${{\tilde{\mathrm{TL}}}}^{b}\leftarrow\Psi_{D}\left(\Psi_{E}\left({(R_{N},z_{N})}_{\mathcal{K}_{i}}^{b};\theta_{E}\right);\theta_{D}\right)$,
${\tilde{\mathrm{TL}}}^{b}\in\mathbb{R}^{N_{b}\times R_{N}\times z_{N}}$;

Calculate loss $\mathcal{L}$ via Eq. (11);

Estimate gradients $\hat{\mathbf{g}}$ using automatic differentiation;

Update parameters: $\theta\leftarrow ADAMW(\hat{\mathbf{g}})$;

end while

end for

Updated parameters: $\\{\theta^{*}:=(\theta_{E}^{*},\theta_{D}^{*})\\}$

Algorithm 1 Range-Dependent Conditional Convolutional Network Training

Once the model is trained, predicting transmission loss becomes
straightforward. The network is used to forecast transmission loss on input
bathymetry meshes using testing data sets. These data sets are created by
segregating some data from the training set. Given that the true and predicted
solutions in this context can be represented in a rasterized format on a
Cartesian grid, the accuracy of the predicted solutions was assessed using the
Structural Similarity Index Measure (SSIM). SSIM is a statistical measure
employed to compare two datasets of the same dimensions with rasterized
representations. Further details on SSIM can be found in Ref. Wang  _et al._
(2004). An SSIM value of 1.0 between two inputs indicates perfect similarity,
while a value of 0 implies no similarity. Next, we turn our attention to a
detailed assessment of the model.

##  IV Test scenario for RC-CAN based learning

###  IV.1 Test problem

To illustrate the learning capability of RC-CAN in predicting ocean acoustic
transmission loss, we examine an oceanic environment with depth-dependent
sound speed following Munk’s sound speed profile Jensen  _et al._ (2011). The
computational domain for the ocean is depicted in Fig. 7. In this scenario,
the domain spans 100 kilometers in range and has a depth of 3000 meters,
featuring a piece-wise linear bathymetry profile. This research aims to employ
a data-driven method to learn transmission loss from 230 Hz point sources at a
depth of 18 meters, with ocean bathymetry varying along the range. We
specifically consider transmission losses primarily attributed to geometric
spreading and reflections from the top and bottom surfaces, excluding losses
due to volume attenuation. The ocean bottom is treated as a rigid boundary,
while the ocean surface is regarded as a fully reflective pressure release
boundary Jensen  _et al._ (2011). Consequently, while reflections occur from
the ocean surface and bottom, scattering losses from the ocean bottom are
deemed negligible.

Figure 7: An illustration of ocean domain with range-dependent bathymetry

###  IV.2 Generation of ground truth for far-field transmission loss

In the physical conditions outlined in this study, the transmission of
underwater radiated noise can be viewed as a linear wave propagation process.
Consequently, it can be mathematically represented in the frequency domain
using the Helmholtz equation in Cartesian coordinates $\bm{\mathrm{x}}$.

$\nabla^{2}p+\frac{\omega^{2}}{c^{2}(\mathbf{x})}p=-\delta\left(\mathbf{x}-\mathbf{x}_{0}\right),\quad\bm{\mathrm{x}}\in\Omega,$
(12)

subject to the boundary conditions. In the context of a two-dimensional ocean
domain $\Omega$, the variable $p$ denotes the far-field acoustic pressure
perturbation originating from a source located at $\bm{\mathrm{x_{0}}}$. We
can express $\bm{\mathrm{x}}$ as $\left(R,z\right)$, and $\nabla^{2}$ as
$\frac{\partial^{2}}{\partial R^{2}}+\frac{\partial^{2}}{\partial z^{2}}$,
which is a valid assumption without loss of generality. Here,
$c\left(\bm{{x}}\right)$ represents the speed of sound, $R$ denotes the
direction along the range of the ocean domain from a reference point, and $z$
indicates the depth from the ocean surface. Equation 12 can be approximately
solved using ray tracing techniques, enabling the analysis of a broad range of
depth and range-dependent variations in ocean environment parameters (e.g.,
sound speed, bathymetry, etc.). The ray method is especially beneficial for
tackling range-dependent problems at high frequencies, situations where normal
mode or parabolic models are not practical options.

While ray tracing techniques ignore the lower frequency solutions of the
Helmholtz equation Jensen  _et al._ (2011), established computational methods
for ray tracing Porter and Bucker (1987); Porter and Liu (1994) have
demonstrated the ability to yield reasonably accurate results even at
frequencies as low as 50 Hz Porter (2010). Ray tracing approaches can cause
anomalies such as sharp caustic and shadow zones in ray amplitude
calculations. However, the use of geometric beams in ray-beam tracing solvers
can mitigate the occurrence of caustics to acceptable levels, as documented in
Ref. Porter and Liu (1994).

In this study, we calculate the far-field acoustic pressure levels considering
geometric spreading, ocean surface, and bottom reflections using BELLHOP
Gaussian beam tracing solver Porter (2010). BELLHOP is employed to compute the
acoustic pressure at evenly spaced receiver locations along the depth ($z$)
and range ($R$) throughout the prescribed ocean domain, for a given point
source depth ($z_{s}$). These acoustic pressure amplitudes are utilized to
derive transmission loss, and a snapshot of the transmission loss distribution
is extracted on a uniformly discretized $R-z$ plane for the ocean domain
depicted in Fig. 7. Our objective is to develop a generalized learning model
for far-field transmission loss in bathymetry-varying scenarios using the RC-
CAN model. This involves training the model on various bathymetry profiles and
demonstrating transmission loss predictions for any bathymetry not included in
the training set.

It’s worth noting that although BELLHOP computes acoustic pressure, we train
the RC-CAN on transmission loss data obtained by post-processing the intensity
of the pressure signal (according to Eq. 5). This choice is made because
acoustic pressure varies over several orders of magnitude in such a large
domain, and deep learning networks cannot be effectively trained on such data.
Conversely, the transmission loss of acoustic pressure is easier to interpret
and more suitable for training the network.

##  V  Results

In this section, we demonstrate the capability of the proposed RC-CAN
architecture to make data-driven predictions of transmission loss,
particularly in far-field scenarios. We also showcase its ability to
generalize across various ocean bathymetry conditions. To validate the
reliability and accuracy of our methodology, we will test it in scenarios
involving the prediction of underwater radiated noise over idealized
seamounts, wedge profiles, and Dickins seamount. Various components of the RC-
CAN architecture were trained with PyTorch Paszke  _et al._ (2019) library.

###  V.1 Training of RC-CAN

The RC-CAN was trained using BELLHOP-generated data to account for
transmission loss distribution across a range of varying bathymetry
conditions. The training process involved sequential training on different
datasets, starting with an ideal seamount dataset, followed by wedge profiles,
and concluding with the Dickins seamount dataset. Fig. 8 illustrates the
progression of mean-squared error during training and validation across these
datasets. Notably, the model required the greatest number of epochs to train
on the dataset with varying sea mount heights. Training patience (i.e., the
number of epochs to wait before early stop if no progress on validation set)
of 100 epochs was set, indicating that if the validation mean-squared error
did not decrease within these epochs, the training process would stop.
Subsequently, the model underwent re-training on a replay buffer dataset
$\mathcal{K}{\text{i}}$ to preserve knowledge from previous tasks. Transfer
learning was applied when transitioning to a new dataset
$\mathcal{D}{\text{i}}$, significantly reducing the required number of epochs
for convergence. For instance, training on a dataset with varying sea mount
bases only necessitated 200 epochs, compared to approximately 300 epochs
needed for the dataset with varying sea mount heights (as depicted in Fig. 8).

((a)) Seamount Height Varying ($\mathcal{D}_{\text{1}}$)

((b)) Seamount Base Varying ($\mathcal{D}_{\text{2}}$)

Figure 8: Training and validation loss plot for dataset
$\mathcal{D}={\mathcal{D}_{\text{1}},\&\mathcal{D}_{\text{2}}}$

###  V.2 Underwater Radiated Noise

The transmission loss distribution computed over the domain via BELLHOP for a
source at a depth $z_{s}=18$ meters is shown in Fig. 9. BELLHOP and most other
routine rays/beam tracing solvers lead to spurious results when the pressure
levels decrease many orders of magnitude Porter (2010); Jensen  _et al._
(2011). Such a rapid drop in pressure levels usually occurs at the boundary of
the shadow zones where no ray traces enter. We observed spurious results as
the pressure levels dropped beyond 10 orders of magnitude. Thus, pressure
levels beyond 10 orders of magnitude were filtered away leading to a flat
region in the transmission loss contour beyond 200 dB in Fig. 9. For obtaining
the data sets we considered Munk’s sound speed profile sampled with 25 depth-
wise velocities. Therefore, the ocean depth was divided into 24 depth-wise
divisions with uniform fluid properties during BELLHOP’s numerical ray
trajectory computations. Notably, the sound speed profile remains constant
across the range, and these assumptions are deliberately chosen to examine the
influence of range-dependent variations in bathymetry on the prediction of
transmission loss.

Figure 9: Transmission loss distribution obtained via BELLHOP for a source
depth of 18 meters over ideal seamount

The underwater radiated noise depends on the geometry of ocean bathymetry. As
the ocean profile changes, the transmission loss at a far-distance location
varies dramatically. The presence of a seamount can significantly impact the
resulting transmission loss in underwater acoustic environments. Seamounts can
act as barriers, causing sound channels to form that can enhance sound
propagation over long distances. Additionally, seamounts can create regions of
quiet or reduced sound levels, particularly in their lee (downwind) sides, due
to sound shadowing effects Ebbeson and Turner (1983). Understanding these
effects is crucial for predicting and interpreting acoustic propagation in
oceanic environments. A neural network trained on underwater radiated noise
data predicts the transmission loss not only for the training set but also for
ocean bathymetry profiles outside of the training range (extrapolation). While
a perfect extrapolation may be challenging in practice, we strive for a small
extrapolation error. Figure 9 below shows the transmission loss estimated from
the bellhop Gaussian beam solver for an ideal sea mount profile.

###  V.3 Far-field transmission loss prediction for ideal seamount

In this section, we evaluate the predictive accuracy of our range-dependent
convolutional neural network framework across various ideal seamount
bathymetry profiles. We examine three distinct test scenarios: Test Case 1
with peak depths at 450 m and 1650 m, Test Case 2 featuring varying base width
with a peak depth of 1200 m and base located at [24 km, 43 km], and Test Case
3 representing a general seamount with a peak depth of 1600 m and base at [12
km, 44 km]. These test cases involve extrapolation beyond the bathymetry range
covered in the training dataset.

Starting with Test Case 1, characterized by a seamount’s peak depth of 1650 m,
Figure 10 showcases a comparison between the simulated transmission loss and
the predictions generated by the range-conditional convolutional neural
network. Notably, our proposed architecture adeptly captures the underwater
radiated noise propagation in this scenario, demonstrating its precision in
modeling complex acoustic phenomena in the deep ocean environment. We measure
the similarity between the prediction and simulation data using SSIM as
mentioned previously. The SSIM in this case is 0.89 demonstrating that the
prediction is 89 percent similar to the simulation data in the extrapolation
regime. The red and warmer region in Figure 10 corresponds to the least
transmission loss and loudest noise region whereas the blue and cooler region
represents maximum transmission loss and the quietest zone in the ocean
domain. The RC-CAN as depicted in Figure 10 correctly predicts the loud zones
and the quiet zones which are important for building the adaptive noise
mitigation strategy in real-time.

Figure 10: Peak depth 1650 m. Top, prediction from RC-CAN, and bottom,
simulated ground truth from BELLHOP solver.

In Test Case 1, where the peak depth is 450m, the structural similarity index
(SSIM) value is 0.91, demonstrating the RC-CAN architecture’s extrapolation
capabilities. To delve deeper into the transmission loss characteristics, we
proceeded to plot the transmission loss at two probe locations, one at a depth
of 500m and another at 1500m. Figure 11 illustrates the variation of
transmission loss with range at these probe depths. The plot indicates that
RC-CAN effectively captures both the region of maximum loss and areas of low
transmission loss. The convolution filter in the RC-CAN architecture
introduces Gaussian smoothing, which filters out high-frequency content, as
seen in Figure 11. As previously stated, there is an excellent level of
agreement between RC-CAN predictions and simulated ground truth across the
entire domain in both cases. However, there is a difference in the zone
beginning at 60 km, where acoustic waves converge to form a loud zone. These
loud zones indicate the convergence of acoustic rays, which contain high-
frequency content that is usually smoothed by the convolution kernel. Despite
this smoothing effect, RC-CAN can correctly identify the locations of these
zones.

((a)) Probe depth = 500m

((b)) Probe depth = 1500m

Figure 11: Transmission loss with range for peak depth of sea mount = 450 m.
(a) Probe depth = 500m, and (b) probe depth = 1500m

For the test case where the base of the ideal seamount varies, with the base
located at 24 km and 43 km and a peak depth of 1200m, the SSIM between RC-CAN
and the simulation ground truth is 0.96. This result is depicted in Figure 12.
Additionally, the transmission loss at depths of 500m and 1500m is analyzed
and presented in Figure 13 to gain further insights. Similar to the height-
varying case, the variation of transmission loss with range is in good
agreement with the simulation results from BELLHOP. This test case illustrates
how the RC-CAN transmission loss prediction changes as the base of the
seamount varies from the source of the sound.

Figure 12: Peak’s depth 1200 m and base at [24km, 43km]. Top, prediction from
RC-CAN, and bottom, simulated ground truth from BELLHOP solver.

((a)) Probe depth = 500m

((b)) Probe depth = 1500m

Figure 13: Transmission loss with range for peak depth of sea mount = 1200 m
and base at [24km, 43km]. (a) Probe depth = 500m, and (b) probe depth = 1500m

In Test Case 3, where both the peak depth and base of the seamount vary, the
SSIM is 0.88. Figure 14 illustrates the variation of transmission loss with
range at a probe depth of 500m. There is a high level of agreement between the
predictions of RC-CAN and the simulated ground truth across the entire domain.
Across all test cases, the results underscore the superior predictive
capabilities of the range-conditional convolutional neural network in
capturing the complex dynamics of underwater noise propagation. The SSIM
values calculated for all test cases are close to or greater than 0.9 in the
extrapolation regime of bathymetry, demonstrating the model’s ability to
generalize across different seamount heights and bases. This positions it as a
robust and advanced solution for accurate transmission loss prediction in
diverse underwater environments. The effectiveness of our proposed
architecture suggests its potential for predictive modeling in underwater
acoustics, making it a valuable tool for marine engineering applications.

((a)) Probe depth = 500m Figure 14: Transmission loss with range for peak
depth of sea mount = 1600 m and base at [12km, 44km].

###  V.4 Far-field transmission loss prediction for wedge profile

The ocean bathymetry features not only seamounts but also gradually sloping
floors. A reliable model should be capable of accurately predicting sound
propagation under these diverse bathymetric conditions. Unlike seamounts, the
gradually sloping floor does not create shadows but rather represents areas
where sound waves combine to form loud zones. Identifying these zones,
especially during the movement of marine vessels, is crucial to minimize their
impact on marine environments.

To evaluate the model’s performance in such scenarios, we designed a test case
where the depth increases from 2180 m to 2380 m at 40 km and then gradually
decreases back to 2100 m at a range of 100 km. The Structural Similarity Index
(SSIM) between the prediction and RC-CAN for this case is 0.98, indicating
that RC-CAN can accurately predict the trajectory of rays in this complex
bathymetric scenario. Figure 15 illustrates the transmission loss profile
prediction by RC-CAN, showing a close alignment with the simulated ground
truth.

Figure 16 further demonstrates the remarkable predictive performance of RC-
CAN, closely aligning with the simulated ground truth across the entire
domain. This test case highlights the model’s capability to accurately predict
sound propagation in scenarios involving gradually sloping floors, showcasing
its robustness in handling diverse underwater environments.

Figure 15: Transmission loss prediction for test case 1. Top, prediction from
RC-CAN, and bottom, simulated ground truth from BELLHOP solver.

((a)) probe depth = 500m

((b)) probe depth = 1500m

Figure 16: Transmission loss prediction for test case 1.

###  V.5 Dickins sea mount

It is crucial to have a model that accurately predicts the transmission loss
profile for realistic ocean bathymetry. Seamounts are common features in the
ocean. An example is Dickins Seamount, located in the Northeast Pacific Ocean
Ebbeson and Turner (1983). To further evaluate the model’s performance for
realistic seamount profiles, we sampled the ocean bathymetry around the
Dickins Seamount location. The Structural Similarity Index (SSIM) for the
Dickins Seamount test case is 0.94, indicating good agreement with the
simulation result. The prediction is illustrated in Figure 17, while the
transmission loss with range is depicted in Figure 18. The RC-CAN model
accurately identifies the location of ray convergence loud zones as well as
the shadow zones.

The results from the Dickins Seamount test case highlight the robustness and
reliability of the RC-CAN model in handling complex bathymetric features. Such
capabilities are essential for accurately predicting transmission loss in
real-world ocean environments, where seamounts and other underwater features
can significantly impact sound propagation.

Figure 17: Transmission loss prediction for Dickins sea mount. Top, prediction
from RC-CAN, and bottom, simulated ground truth from BELLHOP solver. ((a))
probe depth = 500m  Figure 18: Transmission loss prediction for Dickins sea
mount.

The mean SSIM accuracy for transmission loss in the test dataset is 90%.
Additionally, the RC-CAN model is able to compute the transmission loss
distribution over the entire domain for each bathymetry within 0.01-0.5 CPU
seconds. In comparison, similar predictions using BELLHOP required 75-150 CPU
seconds for each bathymetry. Notably, the RC-CAN model is trained on the GPU
and utilizes fast GPU computation for achieving real-time results. This
significant difference in computation time highlights the potential
application of the RC-CAN model for real-time decision-making and control in
underwater acoustic environments.

##  VI  Conclusions

In this article, the authors introduce the range-dependent conditional
convolutional neural network (RC-CAN) for data-driven learning of far-field
acoustic transmission in ocean environments with varying bathymetry. We
propose a replay-based training strategy and range-dependent conditional
convolutional neural network for modeling underwater radiated noise. The model
effectively captures the complex relationship between ocean bathymetry and
transmission loss. By transforming the input mesh into a latent space and
decoding the transmission loss in a single step, the model demonstrates
efficient learning capabilities.

When trained with full-domain snapshots of the transmission loss distribution,
the RC-CAN model showed a generalized prediction capacity of far-field
transmission loss for bathymetry outside the training set. Furthermore, such
predictions were obtained in a physically consistent manner. Specifically, the
RC-CAN model trained with transmission loss distribution obtained via ray/beam
tracing solver Bellhop for idealistic bathymetry like ideal sea mount and
gradually decreasing sea floor can predict transmission loss for Dickins sea
mount with 90% SSIM accuracy. Our empirical findings underscore the
effectiveness of the single-shot learning strategy and the range-conditional
architecture, both of which act as powerful implicit biases within the deep
neural network. This results in enhanced generalization performance across
varying ocean bathymetry conditions. Throughout our study, the proposed model
has consistently demonstrated efficient and accurate prediction of
transmission loss.

We used medium-fidelity ray tracing solvers to get the ground truth for ocean
acoustic transmission loss, which was then used to train and test the RC-CAN
model. It’s important to note that the RC-CAN model is data-driven and
agnostic to how the data is acquired. Therefore, the RC-CAN model can
seamlessly incorporate data obtained from higher-fidelity solvers or
experimental measurements if available. This underscores the potential of
data-driven deep learning models like RC-CAN to generate a digital twin of
ocean acoustic transmission over a wide range of parameters and physical
phenomena of varying complexity. The real-time online prediction capability of
such a digital twin holds promise for fast and physics-informed decision-
making in marine vessel operations. By leveraging the RC-CAN model’s ability
to accurately predict transmission loss profiles in real-time, marine
operators can make informed decisions to optimize vessel operations and
minimize the impact on marine environments.

###### Acknowledgements.

The present study is supported by Mitacs, Transport Canada, and Clear Seas
through the Quiet-Vessel Initiative (QVI) program. We express our gratitude to
Mr. Gaurav Bhatt from the UBC Department of Computer Science for his numerous
invaluable suggestions and discussions regarding the framework for continual
learning. The authors would like to express their gratitude to Dr. Paul
Blomerous and Ms. Tessa Coulthard for their valuable feedback and suggestions.
Additionally, we gratefully acknowledge the contributions of Dr. David Rosen
and Dr. Andrew Trites, who provided insights on the marine biology and
conservation aspects of the study. We would also like to acknowledge that the
GPU facilities at the Compute Canada clusters were used for the training of
our deep learning models.

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