Neural Network

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Period ending 2026-09-21

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658 papers

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May 28, 2026 · cs.LGJ/K move · Enter open · S save

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

Qihong Yang, Yangtao Deng, Qiaolin He, Shiquan Zhang

School of Mathematics, Sichuan University, Chengdu, 610065, China

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).

Neural Partial Differential Equation Solvers Partial Differential Equation Benchmarks Neural Solvers Neural Network Tensor Programs

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