Maximum-likelihood Estimators in Physics-Informed Neural Networks for High-dimensional Inverse Problems

TL;DR

This paper introduces a hyperparameter-free maximum-likelihood estimator framework for physics-informed neural networks, enhancing high-dimensional inverse problem solving by explicit error propagation and SVD-based residual projection.

Contribution

It reformulates inverse PINNs as MLE problems, enabling explicit error propagation and robust solutions without hyperparameter tuning, especially for high-dimensional coupled ODEs.

Findings

MLE formulation allows explicit error propagation.

SVD provides uncorrelated subspaces for residual projection.

Preconditioning with SVD improves robustness in inverse PINNs.

Abstract

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 by differential algebraic equations that are common in transient chemical and biological kinetics. Furthermore, we show that singular-value decomposition (SVD) of the ODE coupling matrices (reaction stoichiometry matrix) provides reduced uncorrelated subspaces in which PINNs solutions can be represented and over which residuals can be projected. Finally, SVD bases serve as preconditioners for the inversion of covariance matrices in this hyperparameter-free robust application of MLE to ``kinetics-informed neural networks''.