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

Area of Science:

• Computational Science
• Applied Mathematics
• Numerical Analysis

Background:

• Solving forward and inverse dynamic problems in 3D space presents significant computational challenges.
• Existing methods like physics-informed neural networks (PINNs) have limitations in terms of convergence speed and accuracy for complex problems.

Purpose of the Study:

• To introduce and evaluate a novel neural network framework, Spectral Integrated Neural Networks (SINNs), for solving dynamic problems.
• To enhance the capability of neural networks in addressing inverse dynamic problems through polynomial basis function expansion.

Main Methods:

• Developed the SINNs framework utilizing spectral integration for temporal discretization.
• Employed fully connected neural networks for spatial domain solutions of partial differential equations.
• Incorporated polynomial basis functions to improve performance on inverse problems.

Main Results:

• SINNs demonstrated effective and accurate solutions for both forward and inverse problems in heat conduction and wave propagation.
• The framework provided precise and stable results for dynamic problems with extended time durations.
• SINNs showed superior performance compared to PINNs, with faster convergence, higher accuracy, and improved efficiency.

Conclusions:

• SINNs represent an effective and accurate computational framework for solving 3D dynamic problems.
• The proposed method offers significant advantages over traditional PINNs for heat conduction and wave propagation.
• SINNs show promise for tackling challenging inverse dynamic problems and long-time simulations.