# Numerically stable neural network for simulating Kardar-Paris-Zhang growth
in the presence of uncorrelated and correlated noises

Tianshu Song  School of Materials Science and Physics, China University of
Mining and Technology, Xuzhou 221116, China  School of Information and Control
Engineering, China University of Mining and Technology, Xuzhou 221116, China
Hui Xia<EMAIL_ADDRESS>School of Materials Science and Physics, China
University of Mining and Technology, Xuzhou 221116, China

###### Abstract

Numerical simulations are important 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 modify 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 for
overcoming 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 estimate effectively the scaling exponents eliminating
numerical divergence.

††preprint: APS/123-QED

##  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 has the form:

$\frac{\partial h(x,t)}{\partial
t}=\nu\nabla^{2}h(x,t)+\frac{\lambda}{2}(\nabla h(x,t))^{2}+\eta(x,t),$ (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 the satisfactory
results have been achieved in terms of both analytical predictions and
numerical simulations. However, there is still inconsistent results for the
KPZ equation in the presence of the correlated noises. When the long-range
spatiotemporal correlations are introduced, the noise satisfies

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

where $\rho$ and $\theta$ are spatial and temporal correlation exponents,
respectively. The KPZ equation with long-range correlations is difficult to be
solved theoretically. Although several theoretical schemes were applied to the
correlated growth system [5, 6, 7, 8, 9, 10], yet these theoretical
predictions exist 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 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
can 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, 19, 16]. 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 there exist evident discrepancies 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.

Figure 1: The proposed framework on simulating stochastic growth system.

In recent years, several neural network-based methods are 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 simulating popular PDEs, such as the Navier-Stokes equations, Schrödinger
equation, and Burgers equation. However, the related researches on dynamic
scaling and numerical divergence of the stochastic PDEs are still lacking,
which motivate us to investigate this interesting issue, and the well-known
KPZ equation becomes our first choice as a typical representive of various
stochastic dynamic system.

The paper is organized as follows: Firstly, we propose a convolutional neural
network-based scheme for simulating the KPZ equation driven by uncorrelated
and correlated noises. Then, we provide the proposed framework, and implement
the detailed training. Next, we exhibit our numerical simulations and compare
them with previous related results. Finally, the corresponding discussions and
conclusions are given.

##  II Model and Method

In this section, we introduce a simple and effective convolutional neural
network (CNN)-based framework to directly simulate KPZ equation. Simulating
stochastic PDEs with the neural network was rarely investigated. Different
from the determinant PDEs (e.g. Schrödinger and Burgers equations), directly
obtaining height field $h(x,t)$ at a specific time and position is impossible
because of the unpredictable stochastic disorder. Therefore, we process the
KPZ equation as a discrete-time model, and build a one dimensional
convolutional neural network $N$ to represent the first two terms of the right
side of Eq. (1):

$N(h)=\nu\nabla^{2}h+\frac{\lambda}{2}(\nabla h)^{2}.$ (3)

Since the neural network works in a data-driven manner, we need to obtain
enough experimental data for training the network. Considering the KPZ
equation with Gaussian white noise could be simulated normally at the early
growth time, we generate enough training data from simulating KPZ equation
within this early growth regions.

Based on training data, the network $N$ can commendably satisfy Eq. (3). Then,
we obtain $h(x,t+1)$ by adopting the discretized scheme,

$h(x,t+1)=h(x,t)+[N(h(x,t))+\eta(x,t))]\Delta t,$ (4)

where $t=1,2,...,T/\Delta t,x=1,2,...,L$. Here, $T$ is the total growth time,
$L$ denotes the system size, and $\Delta t$ is the time step.

It should be noted that, Eq. (3) does not contain the noise term, and the
network only learns to represent the two determinant terms of the KPZ
equation. Therefore, Eq. (4) can be adopted to process different uncorrelated
and correlated noises. By adopting the proposed method, we can directly change
the noise term in Eq. (4) to simulate KPZ equation with both spatially and
temporally correlated noise.

The proposed framework has two advantages. First, without directly
discretizing nonlinear term, it can avoid numerical divergence even at the
long growth time. Second, with the help of convolutional architecture, the
network scheme can directly be adopted to simulate KPZ system in the presence
of various types of noise without adding additional retraining, which is more
universal than the traditional simulation methods.

The proposed framework is shown in Fig. 1. It consists of the training and
growth stages. During the training stage, the network aims to learn the
nonlinear mapping of Eq. (3). The training data of network input X and output
Y can be obtained by

$\left\\{\begin{matrix}\vspace{1ex}&&X_{x,t}=h(x,t),\\\
\vspace{1ex}&&Y_{x,t}=\nu[h(x+1,t)-2h(x,t)+h(x-1,t)]\\\
\vspace{1ex}&&\quad+(\lambda/8)[h(x+1,t)-h(x-1,t)]^{2},\\\
\end{matrix}\right.$ (5)

with periodic boundary condition. In Eq. (5), $h(x,t)$ follows the traditional
finite difference (FD) scheme,

$\left\\{\begin{matrix}\vspace{1ex}&&h(x,0)=0,\\\
\vspace{1ex}&&h(x,t\\!+\\!1)\\!=\\!X_{x,t}+\\![Y_{x,t}+\\!\eta(x,t)]\\!\Delta
t\\!.\\\ \end{matrix}\right.$ (6)

After obtaining the training data, we adopt the back-propagation algorithm to
train the neural network. First, the network predicts
$\textbf{Y}^{\prime}_{x,t}$ from input $\textbf{X}_{x,t}$,

$\textbf{Y}^{\prime}_{x,t}=N(\textbf{X}_{x,t}).$ (7)

Then, the predictions are sent into the loss function $\mathcal{L}$ for
calculating the total error $E$, which reads

$E=\mathcal{L}(\textbf{Y}^{\prime}_{x,t},\textbf{Y}_{x,t}).$ (8)

Finally, the error is back-propagated [23] for modifying parameters of $N$:

$N_{p}=N_{p}-r\frac{\partial E}{\partial N_{p}},$ (9)

where $N_{p}$ denotes the parameters of network $N$, and $r$ is the learning
rate.

During the training process, the network satisfies Eq. (3). Then, we perform
simulations by adopting the commendably trained network within the early
growth regime. It can be observed from Fig. 1 that we send $h(x,t)$ into the
network $N$ and obtain $N(h)$. Then, by introducing the chosen noise, we
obtain $h(x,t+1)$ in accordance with the growth rule from Eq. (4).

###  II.1 Training Details

During the detailed implementations, we first construct a five-layer neural
network, called KPZNet, for simulating KPZ system, whose structure details are
listed in Tab. 1. Then, we obtain the training data $\textbf{X}_{x,t}$ and
$\textbf{Y}_{x,t}$ by discretizing the KPZ equation with standard Gaussian
white noise. The parameters in the KPZ equation we utilized are
$L=128,T=8\times L^{3/2},\Delta t=0.1,\nu=1,\lambda=4$. At each epoch, we
uniformly sample 1/2 number of $i$ for $i<L^{3/2}/\Delta t$ and 1/35 for
$i>L^{3/2}/\Delta t$ to reduce the training data number. To increase the
diversity of training samples, we repeat this process 1600 times. Finally, we
obtain $1.6\times 10^{8}$ training pairs at each epoch for training the neural
network. The loss function consists of two part. The first part is the loss
function for the predicted $\textbf{Y}^{\prime}$ of neural network:

$E_{1}\\!=\frac{1}{2}\sum_{i=1}^{N}\sum_{x=1}^{L}{(\textbf{Y}^{\prime}_{x,t}\\!-\\!\textbf{Y}_{x,t})^{2}}\\!+\\!\sum_{i=1}^{N}\sum_{x=1}^{L}{|\textbf{Y}^{\prime}_{x,t}\\!-\\!\textbf{Y}_{x,t}|},$
(10)

where $N$ is the batchsize during training. The second part of our loss
follows the form,

$E_{2}\\!=\frac{1}{2}\sum_{i=1}^{N}\sum_{x=1}^{L}{(\textbf{Z}^{\prime}_{x,t}\\!-\\!\textbf{Z}_{x,t})^{2}}\\!+\\!\sum_{i=1}^{N}\sum_{x=1}^{L}{|\textbf{Z}^{\prime}_{x,t}\\!-\\!\textbf{Z}_{x,t}|}.$
(11)

Here, $\textbf{Z}^{\prime}_{x,t},\textbf{Z}_{x,t}$ are obtained by

$\left\\{\begin{matrix}\vspace{1.5ex}&&\textbf{Z}^{\prime}_{x,t}\\!=\\!\nu(\textbf{Y}^{\prime}_{x+1,t}\\!+\\!\textbf{Y}^{\prime}_{x-1,t}\\!-\\!2\textbf{Y}^{\prime}_{x,t})\\!+\\!\lambda/2(\textbf{Y}^{\prime}_{x+1,t}\\!-\\!\textbf{Y}^{\prime}_{x,t})^{2},\\\
&&\textbf{Z}_{x,t}\\!=\\!\nu(\textbf{Y}_{x+1,t}\\!+\\!\textbf{Y}_{x-1,t}\\!-\\!2\textbf{Y}_{x,t})\\!+\\!\lambda/2(\textbf{Y}_{x+1,t}\\!-\\!\textbf{Y}_{x,t})^{2}.\\\
\end{matrix}\right.$ (12)

Thus, the total error $E$ is

$E=E_{1}+E_{2}.$ (13)

During training, the batch size is set as $N=32$, and the optimizer we
utilized is Adam [24]. After each epoch, we test the neural network on the
test data $\textbf{X}^{test},\textbf{Y}^{test}$, which are acquired by Eq. (5)
with the same parameters during the training stage. The test error is
monitored to avoid overfitting, and the model with the lowest test error is
saved. The initial learning rate is $r=0.003$, and if the test error does not
decrease for 2 epochs, the learning rate is multiplied by 0.2. When the
learning rate is smaller than $5\times 10^{-6}$, we finish the whole training
process.

Table 1: Network Structure of KPZNet. $N,L$ in the output size respectively means batch size and system size.  Layer | Kernel | Output
---|---|---
Name | (size, channel, padding) | $batch\times channel\times size$
Input | / | $N\times 1\times L$
Conv-1 | (3,3,1) | $N\times 3\times L$
Conv-2 | (3,3,1) | $N\times 3\times L$
Conv-3 | (1,64,0) | $N\times 64\times L$
Conv-4 | (1,3,0) | $N\times 3\times L$
Conv-5 | (3,1,1) | $N\times 1\times L$

###  II.2 Training Results

The errors during training are illustrated in Fig. 2, and it can be observed
that the total errors of both training and test stages quickly decrease and
converge to small values, which means the neural network is well-trained to
represent the discretized KPZ equation without overfitting.

Figure 2:  The errors of the training and test stages.

Figure 3:  The log-log plot of the scaled interface width versus the scaled growth time of the growth system driven by Gaussian white noise: (a) KPZ with scaling exponents $\alpha=0.495$ and $z=1.550$, and (b) EW with $\alpha=0.490$ and $z=1.950$. All results have been averaged over 1000 noise realizations. Table 2: The average divergence time $T_{d}$ for finite difference (FD) and the proposed KPZNet with different $\Delta t$ and noise strength $s$. ’None’ means that the simulation does not diverge during the whole process ($T=10000$).  $\Delta t$ | s |  $T_{d}$ (FD) |  $T_{d}$ (KPZNet)
---|---|---|---
0.01 | 1 | None | None
0.01 | 10 | None | None
0.01 | 100 | 56.8 | None
0.01 | 1000 | 13.4 | None
0.05 | 1 | None | None
0.05 | 10 | 68.7 | None
0.05 | 100 | 10.0 | None
0.05 | 1000 | 6.0 | None
0.10 | 1 | None | None
0.10 | 10 | 17.7 | None
0.10 | 100 | 7.0 | None
0.10 | 1000 | 5.0 | None

To quantitatively describe the scaling properties, we compute the time
evolution of interface width $W(L,t)$ representing the fluctuations of height
field $h(x,t)$, which satisfies the asymptotic scaling properties $W(L,t)\sim
t^{\beta}$ for $t\ll L^{z}$, and $\sim L^{\alpha}$ for $t\gg L^{z}$. Here,
$\beta$, $\alpha$ and $z$ are the growth, dynamic and roughness exponents,
respectively. After training, we adopt the KPZNet to simulate the KPZ equation
by adopting Eq. (4). Figure 3 exhibits the log–log plot of the rescaled
$W(L,t)$ versus the rescaled $t$ for the KPZ growth system driven by Gaussian
white noise. Here, the interface width and growth time are rescaled by
$W/L^{\alpha}$ and $t/L^{z}$, respectively. It’s worth noting that the traning
parameters need to be reasonably chosen to ensure the simulated system into
the true nonlinear KPZ scaling regime. The obtained results are shown in Fig.
3(a). We find that these scaling exponents estimated from the KPZNet are
consistent with the analytical predictions. Therefore, utilizing the neural
network to replace the conventional discretization method is reasonable for
dealing with the KPZ equation driven by the uncorrelated noise.

To prove the advantage that the neural network could gain better control over
numerical stability, we perform simulations by adopting the KPZNet and
conventional FD method of Eq. (6) with different time steps $\Delta
t=0.01,0.05,0.1$ and noise $\eta=s\times N(0,1)$ with different noise strength
$s=1,10,100,1000$, and $N(0,1)$ is the standard Gaussian white noise. We
totally simulate $10000$ steps and record the divergence time $T_{d}$. For
each parameter, we simulate $10$ times and the average $T_{d}$, which is shown
in Tab. 2. We find that the divergence phenomenon could be effectively avoided
by adopting KPZNet. It is worth mentioning that the proposed KPZNet can obtain
accurate scaling exponents with $s=1000$ and $\Delta t=0.05$, which is also
consistent with the previous results [15, 16].

##  III Results and Discussions

Since the KPZNet is trained as an end-to-end paradigm, and the training data
are obtained from directly simulating KPZ equation, one might worry that it
just learns to map $h(x,t)$ obtained from the KPZ universality class. However,
we desire that the KPZNet can be generalized to process the height field with
different distributions. In fact, when $\Delta t$ in Eq. (6) is small enough,
the obtained results could crossover to those of the Edwards-Wilkinson (EW)
class [25], which is equivalent to decreasing the nonlinear coefficient
$\lambda$ to a critcal threshold. Recently, another alternative protocol is
introduced by a small number of the controlled Fourier modes for the crossover
from strong nonlinear to weak coupling in the KPZ growth process [26]. Then we
use a small $\Delta t$ to perform simulations based on the KPZNet, and the
corresponding results obtained are illustrated in Fig. 3(b). In this case, the
scaling exponents obtained from KPZNet are well consistent with the exact
values of the EW universality class. This experiment proves that the KPZNet
can be generalized to process $h(x,t)$ with other distributions of the
training data, indicating that the KPZNet has grasped the growth rule behind
the training data.

To further study the numerical stability and dynamic scaling properties in
dealing with the correlated KPZ system, we adopt the KPZNet to simulate the
KPZ equation driven by long-range temporal correlated noise. The method we
utilized for generating the long-range correlations is the fast fractional
Gaussian noise (FFGN) algorithm [27]. In order to study long-range temporal
correlations in the KPZ growth, previous numerical investigations ususally
adopted discrete growth models with the discrete noise $[0,1]$ or modified the
nonlinear term of the KPZ equation. Here, our proposed KPZNet directly adopt
Eq.(4) to perform numerical simulations. The scaling results of the temporally
correlated KPZ system are obtained, as shown in Fig. 4(a). Here, $\theta=0.30$
is chosen as the typical representative of long-range temporal correlations.
Finally, we perform simulations on the KPZ with spatially correlated noise,
and the simulating results ($\rho=0.30$) are illustrated in Fig. 4(b). Whether
for temporally or for spatially correlated KPZ system, the scaling exponents
are estimated correspondingly based on the good data collapses.

Considering that numerical divergence is happened frequently based on common
simulation method for the correlated noises without specified treatment. In
contrast, Figures. 4(a)-(b) exhibit that, by utilizing the neural network, the
proposed method can grow to the saturation stage without numerical divergence,
and the scaling exponents obtained from correlated noise are consistent with
that of previous results within the error range [15, 16]. It should be noted
that the desired results are achieved by directly adopting Eq.(4) without any
extra process (e.g. changing the amplitude of noise or other parameters),
indicating the high availability of the KPZNet for temporally and spatially
correlated noises.

It can be concluded from all above experiments that the proposed KPZNet
achieves effective scaling exponents for KPZ growth driven by uncorrelated and
correlated noises. The neural network could reveal the main rules behind the
training data, and without additional adjustments, it could also apply to deal
with the linear and nonlinear growth system driven by various types of noises.
Thus, adopting the proposed numerical stable neural network is a promising
approach to perform numerically various types of stochastic dynamic system.

Figure 4:  The collapse results of scaled interface width versus the scaled
growth time from KPZ driven by (a) long-range temporal correlations
($\theta=0.30$) with $\alpha=0.745$ and $z=1.370$, and long-range spatial
correlations ($\rho=0.30$) with $\alpha=0.610$ and $z=1.520$. All data were
averaged over 500 independent noise realizations.

##  IV Conclusions

To summarize, we proposed a numerically stable neural network to simulate the
KPZ equation driven by uncorrelated and correlated noises, for avoiding
numerical divergence and obtaining the expected scaling exponents. We trained
the KPZNet to represent two determined terms of the KPZ equation, and then by
introducing Gaussian white noise to the neural network. The experimental
results prove that the KPZNet has rather better numerical stability than the
commonly used numerical methods. We also adopted the KPZNet to simulate the
KPZ equation driven by both long-range temporal and spatial correlations, and
the estimated scaling exponents could be achieved effectively.

This work makes an exploratory attempt in the direction of utilizing the
neural network to simulate stochastic PDEs. The proposed KPZNet is very robust
for dealing with the chosen KPZ system behind the training data, and it could
be generalized to process interface growth with different distributions. Once
the KPZNet is commendably trained, it can be adopted to various random
environments eliminating numerical divergence, which is the hard task for
traditional simulation methods. We hope that the neural network-based
approaches show better prospects in the stochastic PDEs simulations.

###### Acknowledgements.

We would like to thank Yancheng Wang for many useful discussions and the
critical reading of the manuscript. This work is supported by Key Academic
Discipline Project of China University of Mining and Technology under Grant
No. 2022WLXK04.

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