统一分析物理神经网络泛化能力，揭示非线性对泛化边界的指数影响。
Unified generalization analysis for physics informed neural networks
- 用泰勒展开将非线性微分算子转为高维空间线性算子，统一分析框架
- 证明高秩网络在含微分算子场景下仍具良好泛化性
- 发现微分算子非线性程度越高，泛化边界越宽，影响显著
物理信息神经网络（PINNs）及其变体（VPINNs）通过融入物理定律，适用于科学计算问题。现有针对PINNs和VPINNs的泛化分析受限，常依赖稳定性或线性椭圆性等强假设。本文在统一框架下推导了包含输入变量微分的神经网络泛化界，通过泰勒展开将非线性微分算子表示为高维空间中的线性算子，结合Koopman分析方法，表明即使在涉及微分算子的情形下，高秩网络仍可实现良好泛化。同时发现，微分算子的非线性程度会指数级放大泛化界，凸显其对泛化性能的关键影响。
原文摘要 · Abstract (English)
Physics-Informed Neural Networks (PINNs) and their variational counterparts (VPINNs) are neural networks that incorporate physical laws, making them useful for scientific problems. Existing generalization analyses for PINNs and VPINNs remain limited, often requiring restrictive assumptions such as stability conditions or linear ellipticity. In this paper, we derive generalization bounds for neural networks that involve differentiation with respect to input variables, covering PINNs and VPINNs under a unified framework. We apply Taylor expansion to represent nonlinear differential operators as linear operators on a high-dimensional space, enabling the use of Koopman-based analysis and showing that high-rank networks can generalize well even in settings involving differential operators. We also show that the nonlinearity of the differential operator exponentially enlarges the bound, highlighting its significant impact on generalization.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。