用张量积结构直接求解偏微分方程，无需迭代训练
A Novel Tensor Product-Based Neural Network for Solving Partial Differential Equations
- 通过张量积构造多维基函数，大幅降低参数量
- 线性求解系数，训练速度比传统方法快得多
- 适合长时模拟和非线性问题，精度更高
本文提出张量积神经网络（TPNet），一种用于高效准确函数逼近与偏微分方程求解的新架构。核心思想是将解显式表示为基函数的线性组合，系数通过直接最小二乘法求解，从而跳过传统的梯度训练。主要贡献包括：(1) 高效张量积方案，从两组子网络输出生成多维基函数，显著降低模型复杂度与参数量，同时保持表达能力；(2) 块时间推进策略，提升长时间模拟的计算效率；(3) 非线性项作为源项处理的线性重构策略。实验表明，TPNet在精度和训练时间上均优于主流神经网络求解器。性能优势源于其结构化设计与确定性最小二乘拟合，区别于传统PINN等方法所需的迭代优化。
原文摘要 · Abstract (English)
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).
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