AI 中文总结
该研究将因子分解机衍生的特征交互模块嵌入物理信息神经网络与神经算子，提出FM-PINN、FM-Operator等模型，提升了激波主导等问题的PDE解近似精度，为相关物理建模提供了新方向。
AI 中文摘要
本研究将从因子分解机（Factorization Machines, FMs）衍生的特征交互模块嵌入物理信息神经网络（Physics-Informed Neural Networks, PINNs）与神经算子学习中，以提升模型对参数化偏微分方程（Partial Differential Equations, PDEs）解流形的表达能力。受多元函数二阶泰勒展开表征变量耦合的启发，我们首先提出FM-PINN，该模型可显式捕捉时空变量交互，提升对光滑高阶PDE的近似精度。我们进一步将空间坐标、时间、物理参数及初边值条件划分为独立特征集，建模跨组交互；基于此策略开发的FM-Operator与FM-DeepONet，对非线性守恒律及存在陡峭梯度或不连续性的问题尤为有效，但在光滑算子学习基准上未展现一致优势。数值测试表明，所提机制在挑战性激波主导方程上实现了显著精度提升，为具有强跨场依赖的参数化PDE的物理一致建模提供了有前景的方向。
英文摘要
Despite the many variants of DeepONet that have been proposed, query-based operator networks still struggle with shock-dominated and low-viscosity PDEs, whose sharp moving discontinuities and slowly decaying solution spectra challenge finite-dimensional separable representations. In this work, we propose \emph{Feature Interaction Modeling Operator} (FM-Operator), a point-wise query neural operator that explicitly models feature construction and interactions between sensor observations and query coordinates. Our design is motivated by a reinterpretation of the canonical DeepONet aggregation through the lens of multiplicative interactions. Specifically, the branch--trunk inner product admits the equivalent form \(\boldsymbol{b}(u)^\top \boldsymbolτ(y)=\boldsymbol{1}^\top \operatorname{diag}(\boldsymbol{b}(u))\,\boldsymbolτ(y)\), revealing that the two representations interact only along corresponding latent dimensions and therefore constitute a diagonally constrained multiplicative interaction. This observation suggests that, beyond improving the individual branch and trunk networks, the structure through which function and query representations interact is itself an important inductive bias in point-wise operator learning. FM-Operator accordingly redesigns both feature construction and feature interaction, enabling structured information exchange beyond the conventional branch--trunk coupling while retaining point-wise query evaluation. Experiments across multiple PDE benchmarks demonstrate that FM-Operator consistently outperforms vanilla DeepONet and achieves clear improvements over the strong Shift-DeepONet baseline. These results suggest that explicitly designing representation construction and interaction provides a promising direction for improving the effectiveness of DeepONet-style query-based neural operators.
Comments16 pages
