Maximum-likelihood Estimators in Physics-Informed Neural Networks for High-dimensional Inverse Problems. (arXiv:2304.05991v2 [cs.LG] UPDATED)

Physics-informed neural networks (PINNs) have proven a suitable mathematical
scaffold for solving inverse ordinary (ODE) and partial differential equations
(PDE). Typical inverse PINNs are formulated as soft-constrained multi-objective
optimization problems with several hyperparameters. In this work, we
demonstrate that inverse PINNs can be framed in terms of maximum-likelihood
estimators (MLE) to allow explicit error propagation from interpolation to the
physical model space through Taylor expansion, without the need of
hyperparameter tuning. We explore its application to high-dimensional coupled
ODEs constrained …

application arxiv error expansion hyperparameter likelihood maximum-likelihood mle networks neural networks optimization ordinary physics physics-informed space taylor terms work