Uncertainty-Aware Diagnostics for Physics-Informed Machine Learning
This addresses a critical reliability issue for researchers and practitioners using PIML in scientific and engineering domains, though it is an incremental improvement within existing Gaussian process frameworks.
The paper tackles the problem of ambiguous model quality measurement in physics-informed machine learning (PIML) due to multi-objective training, which can lead to failure modes despite strong statistical metrics. It introduces the Physics-Informed Log Evidence (PILE) score, a single uncertainty-aware metric that effectively selects hyperparameters like kernel bandwidth and regularization weights, and identifies well-adapted kernel choices even before data acquisition.
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.