Physics-informed Deep Learning Based on the Finite Difference Method for Efficient and Accurate Numerical Solution of Partial Differential Equations

Jiaming Zhang, Dirk Husmeier, Hao Gao, David Dalton

Abstract: In this paper, we present important advancements in the application of Physics-Informed Neural Networks (PINNs) for solving complex partial differential equations (PDEs), which are pivotal in modelling phenomena across various scientific and engineering disciplines. Our approach integrates fourth-order Runge-Kutta (RK4) methods into the loss functions of PINNs, to improve solution accuracy in various benchmark problems such as the 1-D Korteweg–de Vries, 2-D Burgers' and the Navier-Stokes equations. Furthermore, we combine a modified Multi-layer Perceptron (MLP) architecture that noticeably improves the predictive accuracy of PINNs over automatic differentiation (AD) based methods while incurring only a minimal increase in computational costs. Through numerical experiments, our findings demonstrate that the proposed RK4-based loss function and modified MLP architecture offer substantial improvements over AD-based methods, particularly in scenarios characterised by nonlinear and high-dimensional PDEs. This study not only bridges the gap between conventional numerical simulations and deep learning approaches for solving PDEs but also opens new avenues for future research in computational physics and engineering. Our contributions promise to enhance the robustness, efficiency, and applicability of PINNs in tackling a broader range of complex physical problems, marking a significant step forward in the intersection of physical sciences, machine learning and statistics.