Model Selection in Physics-Informed Machine Learning
Max Daniels, Massachusetts Institute of Technology
Physics Informed Neural Networks (PINNs) are a recently popularized machine learning paradigm for approximating the full solution of a PDE given observations of the value of the solution at some points in the interior of the domain. The success of this approach depends heavily of the choice of network architecture and it is difficult (even heuristically) to obtain error bounds on a PINN solution. Inspired by the recent Interpolating Information Criterion (IIC), a performance metric for neural network regression models, we introduce an information criterion for a variant of PINNs which uses kernel regression in place of neural networks. This criterion is theoretically motivated by its relationship to PAC-Bayesian generalization error bounds, and it can be used to perform model selection (i.e. to judge the quality of a chosen "architecture&" in terms of its generalization error in a given learning problem).