An Improved Method for Physics-Informed Neural Networks That Accelerates Convergence

Abstract

Physics-Informed Neural Networks (PINNs) have proven highly effective for solving high-dimensional Partial Differential Equations (PDEs), having demonstrated tremendous potential in a variety of challenging scenarios. However, traditional PINNs (vanilla PINNs), typically based on fully connected neural networks (FCNN), often face issues with convergence and parameter redundancy. This paper proposes a novel approach that utilizes a multi-input residual network, incorporating a multi-step training paradigm to facilitate unsupervised training. This improved method, which we named MultiInNet PINNs, can enhance the convergence speed and the stability of traditional PINNs. Our experiments demonstrate that MultiInNet PINNs achieve better convergence with fewer parameters than other networks like FCNN, ResNet, and UNet. Specifically, the multi-step training increases convergence speed by approximately 45%, while the MultiInNet enhancement contributes an additional 50%, leading to a total improvement of about 70%. This accelerated convergence speed allows PINNs to lower computational costs by achieving faster convergence. Moreover, our MultiInNet PINNs provides a potential method for handling initial and boundary conditions (I/BCs) separately within PINNs.

Keywords

• Physics-informed neural network
• partial differential equations
• multi-input residual network
• convergence speed
• unsupervised learning