Unified generalization analysis for physics informed neural networks
Provides theoretical guarantees for PINNs, addressing a key gap in their generalization analysis for scientific machine learning.
The paper derives unified generalization bounds for PINNs and VPINNs without restrictive assumptions like stability or linear ellipticity, showing that high-rank networks generalize well and that nonlinear differential operators exponentially enlarge the bound.
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.