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

arXiv: https://arxiv.org/abs/2304.05991 · HTML: https://mlsanity.com/paper/2304.05991/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 72 references that are in the map.
- First posted on arXiv 12 Apr 2023.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems](https://mlsanity.com/paper/2107.12146/) (2021)
- [Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks](https://mlsanity.com/paper/2008.10653/) (2020)
- [Inverse-Dirichlet Weighting Enables Reliable Training of Physics Informed Neural Networks](https://mlsanity.com/paper/2107.00940/) (2021)
- [Physics Informed Neural Networks for Simulating Radiative Transfer](https://mlsanity.com/paper/2009.13291/) (2020)
- [Autonomous Discovery of Unknown Reaction Pathways from Data by Chemical Reaction Neural Network](https://mlsanity.com/paper/2002.09062/) (2020)
- [The ALAMO approach to machine learning](https://mlsanity.com/paper/1705.10918/) (2017)
- [Autonomous Kinetic Modeling of Biomass Pyrolysis using Chemical Reaction Neural Networks](https://mlsanity.com/paper/2105.11397/) (2021)
- [Kinetics-Informed Neural Networks](https://mlsanity.com/paper/2011.14473/) (2020)
- [The Automated Discovery of Kinetic Rate Models -- Methodological Frameworks](https://mlsanity.com/paper/2301.11356/) (2023)

## Built on by (most cited first, 1 of 1)

- [PBPK-iPINNs: Inverse Physics-Informed Neural Networks for Physiologically Based Pharmacokinetic Brain Models](https://mlsanity.com/paper/2509.12666/) (2025)

## Abstract (arXiv)

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''.