Maximum-likelihood Estimators in Physics-Informed Neural Networks for High-dimensional Inverse Problems
This work addresses hyperparameter tuning challenges in PINNs for inverse problems in kinetics, offering a robust method for researchers in computational chemistry and biology, though it is incremental as it builds on existing PINN frameworks.
The authors tackled the problem of hyperparameter tuning in physics-informed neural networks (PINNs) for high-dimensional inverse problems by framing them as maximum-likelihood estimators, enabling explicit error propagation without hyperparameters. They applied this to coupled ODEs in chemical and biological kinetics, using singular-value decomposition to reduce subspaces and precondition covariance matrices.
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''.