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

TL;DR

This paper introduces a neural network-based simulation method for the KPZ equation that overcomes numerical divergence issues, accurately estimating scaling exponents under various noise conditions, thus improving the reliability of nonlinear growth modeling.

Contribution

A convolutional neural network approach is proposed to simulate the KPZ equation, ensuring numerical stability and accurate scaling exponent estimation in the presence of correlated noises.

Findings

Neural network effectively estimates KPZ scaling exponents.

Method maintains stability with correlated and uncorrelated noises.

Outperforms traditional simulation techniques in stability and accuracy.

Abstract

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.