Uncertainty-Regularized Neural Surrogates for Sparse Governing-Equation Identification from Observations

Abstract

The automated identification of governing equations from observational data has emerged as a transformative paradigm in scientific machine learning, enabling the discovery of complex physical laws without relying on first principles. However, contemporary sparse regression techniques frequently fail when applied to real-world datasets characterized by high levels of measurement noise and sparsity. To address these limitations, this paper introduces a novel framework utilizing uncertainty-regularized neural surrogates for the robust identification of nonlinear dynamics. By embedding probabilistic uncertainty quantification directly into the optimization objective of a neural surrogate model, the proposed method actively filters epistemic and aleatoric noise, yielding highly accurate derivative approximations. These refined spatial and temporal derivatives are subsequently passed to a sparse optimization module that identifies the minimal, most descriptive set of physical terms comprising the underlying partial differential equation. Extensive theoretical analysis and empirical evaluations across several chaotic and fluid dynamical systems demonstrate that the inclusion of an uncertainty-regularization term significantly mitigates the propagation of observational errors. The proposed framework maintains structural recovery accuracy even under severe noise perturbations that typically break deterministic models. This research provides a mathematically rigorous foundation for discovering parsimonious physical laws in highly uncertain operational environments.