Numerically stable neural network for simulating Kardar-Parisi-Zhang
  growth in the presence of uncorrelated and correlated noises
Numerical simulations are essential tools for exploring the dynamic scaling
properties of the nonlinear Kadar-Parisi-Zhang (KPZ) equation. Yet the inherent
nonlinearity frequently causes numerical divergence within the strong-coupling
regime using conventional simulation methods. To sustain the numerical
stability, previous works either utilized discrete growth models belonging to
the KPZ universality class or modified the original nonlinear term by the
designed specified operators. However, recent studies revealed that these
strategies could cause abnormal results. Motivated by the above-mentioned
facts, we propose a convolutional neural network-based method to simulate the
KPZ equation driven by uncorrelated and correlated noises, aiming to overcome
the challenge of numerical divergence, and obtaining reliable scaling
exponents. We first train the neural network to represent the determinant terms
of the KPZ equation in a data-driven manner. Then, we perform simulations for
the KPZ equation with various types of temporally and spatially correlated
noises. The experimental results demonstrate that our neural network could
effectively estimate the scaling exponents eliminating numerical divergence.
Tianshu Song, Hui Xia
Numerically stable neural network for simulating Kardar-Paris-Zhang growth in the presence of uncorrelated and correlated noises

###### Abstract

Numerical simulations are essential tools for exploring the dynamic scaling properties of the nonlinear Kadar-Paris-Zhang (KPZ) equation. Yet the inherent nonlinearity frequently causes numerical divergence within the strong-coupling regime using conventional simulation methods. To sustain the numerical stability, previous works either utilized discrete growth models belonging to the KPZ universality class or modified the original nonlinear term by the designed specified operators. However, recent studies revealed that these strategies could cause abnormal results. Motivated by the above-mentioned facts, we propose a convolutional neural network-based method to simulate the KPZ equation driven by uncorrelated and correlated noises, aiming to overcome the challenge of numerical divergence, and obtaining reliable scaling exponents. We first train the neural network to represent the determinant terms of the KPZ equation in a data-driven manner. Then, we perform simulations for the KPZ equation with various types of temporally and spatially correlated noises. The experimental results demonstrate that our neural network could effectively estimate the scaling exponents eliminating numerical divergence.

## I Introduction

The Kadar-Paris-Zhang (KPZ) equation [1] not only achieves remarkable success in the field of surface growth but also plays a vital role in describing other physical phenomena, such as Bose gas [2], particle transport [3], and stirred fluids [4]. The KPZ equation reads

\[\frac{\partial h(x,t)}{\partial t}=\nu\nabla^{2}h(x,t)+\frac{\lambda}{2}( \nabla h(x,t))^{2}+\eta(x,t), \tag{1}\]

where \(h(x,t)\) is the growth height at position \(x\) and time \(t\), \(\nu\) is the diffusion constant, \(\lambda\) is the coefficient of the nonlinear term characterizing lateral growth, and usually \(\eta(x,t)\) is Gaussian white noise. The universal behaviour and scaling exponents of the KPZ equation driven by this type of noise have been fully explored, and satisfactory results have been achieved in terms of both analytical predictions and numerical simulations. However, there are still inconsistent results for the KPZ equation in the presence of correlated noises. When long-range spatiotemporal correlations are introduced, the noise satisfies

\[\left\langle\eta(x,t)\eta\left(x^{\prime},t^{\prime}\right)\right\rangle\sim \left|x-x^{\prime}\right|^{2\rho-1}\left|t-t^{\prime}\right|^{2\theta-1}, \tag{2}\]

where \(\rho\) and \(\theta\) are spatial and temporal correlation exponents, respectively. The KPZ equation with long-range correlations is challenging to be solved theoretically. Although several theoretical schemes were applied to the correlated growth system [5; 6; 7; 8; 9; 10], these theoretical predictions have some conflicting results.

Meanwhile, previous works [11; 12; 13; 14; 15; 16] performed numerical simulations to explore the scaling properties of the KPZ equation with spatial and temporal correlations. Yet the enormous challenge one must be faced is that simulating the KPZ equation has annoyed numerical instability due to the inherent nonlinearity, which could lead to abnormal growth termination even at the early growth times. In order to avoid this kind of numerical overflow, previous studies typically designed a few special operators [17; 18] to replace the nonlinear term in Eq. (1). It was believed that these operators could hold the universality class unchanged. However, recent works indicated that this strategy for dealing with the inherent nonlinearity may notably cause different universality class [14; 15; 16; 19]. Some other works utilized discrete growth models, _e.g._ ballistic deposition (BD) model [11; 14], to represent the KPZ equation for avoiding numerical divergence. However, recent research revealed that evident discrepancies exist between the BD model and the KPZ equation when long-range correlated noise is introduced [15]. Consequently, replacing the nonlinear term and utilizing discrete growth models are suboptimal for obtaining convincing results in the presence of long-range correlations.

In recent years, several neural network-based methods have been successively proposed to simulate partial differential equations (PDEs). For example, Raissi _et al._[20] proposed a physics-informed neural network to realize the data-driven solution and data-driven discovery of PDEs. Kochkov _et al._[21] adopted a neural network to approximate the Navier-Stokes equations, which achieved 40-80x fold computational speedups for modeling the two-dimensional turbulent flows. Bar-Sinai _et al._[22] employed a neural network to estimate spatial derivatives to satisfy the PDEs, and the network achieved highly accurate solutions with an unprecedentedly low resolution. The above neural network-based research mainly explored the effectiveness of neural networks for simulat