Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs

Abstract

Physics informed neural networks (PINNs) have recently been very successfully applied for efficiently approximating inverse problems for PDEs. We focus on a particular class of inverse problems, the so-called data assimilation or unique continuation problems, and prove rigorous estimates on the generalization error of PINNs approximating them. An abstract framework is presented and conditional stability estimates for the underlying inverse problem are employed to derive the estimate on the PINN generalization error, providing rigorous justification for the use of PINNs in this context. The abstract framework is illustrated with examples of four prototypical linear PDEs. Numerical experiments, validating the proposed theory, are also presented.

References in corpus (8)

• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data
• Adaptive activation functions accelerate convergence in deep and physics-informed neural networks
• fPINNs: Fractional Physics-Informed Neural Networks
• Deep learning observables in computational fluid dynamics
• Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks
• Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data
• A finite element data assimilation method for the wave equation