# Uncertainty-Aware Diagnostics for Physics-Informed Machine Learning

arXiv: https://arxiv.org/abs/2510.26121 · HTML: https://mlsanity.com/paper/2510.26121/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 51 references that are in the map.
- First posted on arXiv 30 Oct 2025.

## Builds on (most cited first, 27 of 27)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Neural Processes](https://mlsanity.com/paper/1807.01622/) (2018)
- [Attentive Neural Processes](https://mlsanity.com/paper/1901.05761/) (2019)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Fast Finite Width Neural Tangent Kernel](https://mlsanity.com/paper/2206.08720/) (2022)
- [Learning Physical Models that Can Respect Conservation Laws](https://mlsanity.com/paper/2302.11002/) (2023)
- [Bayesian Model Selection, the Marginal Likelihood, and Generalization](https://mlsanity.com/paper/2202.11678/) (2022)
- [Determinantal Point Processes in Randomized Numerical Linear Algebra](https://mlsanity.com/paper/2005.03185/) (2020)
- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [Modeling and interpolation of the ambient magnetic field by Gaussian processes](https://mlsanity.com/paper/1509.04634/) (2015)
- [Learning continuous models for continuous physics](https://mlsanity.com/paper/2202.08494/) (2022)
- [A Bayesian information criterion for singular models](https://mlsanity.com/paper/1309.0911/) (2013)
- [Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients](https://mlsanity.com/paper/2212.14319/) (2022)
- [Learning curves for Gaussian process regression with power-law priors and targets](https://mlsanity.com/paper/2110.12231/) (2021)
- [Physics-informed machine learning as a kernel method](https://mlsanity.com/paper/2402.07514/) (2024)
- [The Interpolating Information Criterion for Overparameterized Models](https://mlsanity.com/paper/2307.07785/) (2023)
- [Monotonicity and Double Descent in Uncertainty Estimation with Gaussian Processes](https://mlsanity.com/paper/2210.07612/) (2022)
- [Physics-informed kernel learning](https://mlsanity.com/paper/2409.13786/) (2024)
- [Neural equilibria for long-term prediction of nonlinear conservation laws](https://mlsanity.com/paper/2501.06933/) (2025)
- [The False Promise of Zero-Shot Super-Resolution in Machine-Learned Operators](https://mlsanity.com/paper/2510.06646/) (2025)
- [Determinant Estimation under Memory Constraints and Neural Scaling Laws](https://mlsanity.com/paper/2503.04424/) (2025)

## Built on by (most cited first, 3 of 3)

- [Smoothness Errors in Dynamics Models and How to Avoid Them](https://mlsanity.com/paper/2602.05352/) (2026)
- [FluidGaussian: Propagating Simulation-Based Uncertainty Toward Functionally-Intelligent 3D Reconstruction](https://mlsanity.com/paper/2603.21356/) (2026)
- [A unified perspective of Gaussian process approximation for differential equations](https://mlsanity.com/paper/2607.06292/) (2026)

## Abstract (arXiv)

Physics-informed machine learning (PIML) integrates prior physical information, often in the form of differential equation constraints, into the process of fitting machine learning models to physical data. Popular PIML approaches, including neural operators, physics-informed neural networks, neural ordinary differential equations, and neural discrete equilibria, are typically fit to objectives that simultaneously include both data and physical constraints. However, the multi-objective nature of this approach creates ambiguity in the measurement of model quality. This is related to a poor understanding of epistemic uncertainty, and it can lead to surprising failure modes, even when existing statistical metrics suggest strong fits. Working within a Gaussian process regression framework, we introduce the Physics-Informed Log Evidence (PILE) score. Bypassing the ambiguities of test losses, the PILE score is a single, uncertainty-aware metric that provides a selection principle for hyperparameters of a PIML model. We show that PILE minimization yields excellent choices for a wide variety of model parameters, including kernel bandwidth, least squares regularization weights, and even kernel function selection. We also show that, even prior to data acquisition, a special 'data-free' case of the PILE score identifies a priori kernel choices that are 'well-adapted' to a given PDE. Beyond the kernel setting, we anticipate that the PILE score can be extended to PIML at large, and we outline approaches to do so.