Spectral integrated neural networks (SINNs) for solving forward and inverse dynamic problems

TL;DR

This paper introduces spectral integrated neural networks (SINNs), a new framework combining spectral methods and neural networks to efficiently solve complex forward and inverse dynamic PDE problems with high accuracy and stability.

Contribution

The paper presents a novel spectral integration-based neural network framework that improves convergence, accuracy, and stability in solving dynamic PDEs, outperforming existing physics-informed neural networks.

Findings

SINNs achieve faster convergence than PINNs.

SINNs provide higher computational accuracy.

SINNs demonstrate stable solutions for long-time dynamics.

Abstract

This paper proposes a novel neural network framework, denoted as spectral integrated neural networks (SINNs), for resolving three-dimensional forward and inverse dynamic problems. In the SINNs, the spectral integration method is applied to perform temporal discretization, and then a fully connected neural network is adopted to solve resulting partial differential equations (PDEs) in the spatial domain. Specifically, spatial coordinates are employed as inputs in the network architecture, and the output layer is configured with multiple outputs, each dedicated to approximating solutions at different time instances characterized by Gaussian points used in the spectral method. By leveraging the automatic differentiation technique and spectral integration scheme, the SINNs minimize the loss function, constructed based on the governing PDEs and boundary conditions, to obtain solutions for dynamic problems. Additionally, we utilize polynomial basis functions to expand the unknown function, aiming to enhance the performance of SINNs in addressing inverse problems. The conceived framework is tested on six forward and inverse dynamic problems, involving nonlinear PDEs. Numerical results demonstrate the superior performance of SINNs over the popularly used physics-informed neural networks in terms of convergence speed, computational accuracy and efficiency. It is also noteworthy that the SINNs exhibit the capability to deliver accurate and stable solutions for long-time dynamic problems.