A Novel Tensor Product-Based Neural Network for Solving Partial Differential Equations

2026-05-28

AI Summary

This work addresses the challenges of conventional neural network approaches for solving partial differential equations (PDEs)—notably lengthy training times, optimization difficulties, and parameter redundancy—by introducing the Tensor Product Network (TPNet). TPNet explicitly represents the solution as a linear combination of tensor-product basis functions, with coefficients determined directly via least squares, thereby circumventing iterative optimization. To handle long-time evolution, a block-wise time-marching strategy is employed, while a linear reconstruction mechanism for nonlinear terms enhances numerical stability. The proposed method substantially reduces model parameters without sacrificing expressive power, achieving superior accuracy and computational efficiency compared to existing neural solvers such as Physics-Informed Neural Networks (PINNs), thus enabling fast and stable PDE simulations.

Abstract

This paper presents the Tensor Product Network (TPNet), a novel neural architecture for efficient and accurate function approximation and PDE solving. The core of the proposal involves constructing the solution explicitly as a linear combination of basis functions integrated into the network, with coefficients determined by a direct least-squares solve, thereby bypassing traditional gradient-based training. The key methodological contribution include: (1) an efficient tensor-product scheme that generates multi-dimensional basis functions from combinations of two sets of subnetwork outputs, significantly reducing model complexity and parameter count while maintaining expressivity; (2) a block time-marching strategy to improve computational efficiency in long-time simulations; and (3) a linear reformulation strategy for handling nonlinear PDEs by treating known nonlinear terms as sources. TPNet achieves superior accuracy and shorter training times than conventional neural network solvers. This performance gain stems from its structured design and deterministic least-squares fitting, which contrast with the iterative, often computationally intensive optimization required by mainstream methods like Physics-Informed Neural Networks (PINNs).

Problem

• Partial Differential Equations
• Neural Networks
• Function Approximation
• Computational Efficiency
• Nonlinear PDEs

Innovation

• Tensor Product Network
• Least-squares fitting
• Basis function approximation
• Time-marching strategy
• Nonlinear PDE linearization