将因子分解机引入物理信息神经网络，提升复杂方程求解精度
Feature Interaction Modeling for Physics-Informed Neural Networks and Neural Operators
- 用因子分解机显式建模变量间交互关系
- 在含激波的非线性方程上准确率显著提升
- 适合强耦合、突变性强的物理问题建模
本文将因子分解机（FMs）导出的特征交互模块嵌入物理信息神经网络（PINNs）和神经算子学习中，以增强对参数化偏微分方程（PDEs）解流形的表达能力。受多元函数二阶泰勒展开启发，提出FM-PINN，显式捕捉时空变量间的交互，提升光滑高阶PDE的逼近精度。进一步将空间坐标、时间、物理参数及初边值条件分组，建模跨组交互，发展出FM-Operator与FM-DeepONet，对非线性守恒律及具有尖锐梯度或不连续性的难题尤为有效，但在平滑算子学习任务上无一致优势。数值实验表明，该机制在挑战性激波主导方程上带来显著精度提升，为强跨场依赖的参数化PDE物理一致性建模提供了新方向。
原文摘要 · Abstract (English)
This work embeds feature interaction modules derived from factorization machines (FMs) into physics-informed neural networks (PINNs) and neural operator learning, to enhance model expressiveness for solution manifolds of parameterized partial differential equations (PDEs). Motivated by the second-order Taylor expansion of multivariate functions to characterize variable couplings, we first propose FM-PINN. It explicitly captures spatio-temporal variable interactions and improves the approximation accuracy for smooth high-order PDEs. We further group spatial coordinates, time, physical parameters, and initial and boundary conditions into independent feature sets and model their cross-group interactions. Based on this strategy, we develop FM-Operator and FM-DeepONet, which are particularly effective for nonlinear conservation laws and problems with sharp gradients or discontinuities, while offering no consistent advantage on smooth operator learning benchmarks. Numerical tests demonstrate that the proposed mechanism delivers substantial accuracy gains on challenging shock-dominated equations, indicating a promising direction for physics-consistent modeling of parameterized PDEs with strong cross-field dependencies.
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