S-PENNs Quantify Uncertainty in Physics-Constrained ML Models

Scientific machine learning models often embed physical structures directly into their architectures to ensure consistency with known laws. However, quantifying uncertainty in these "hard-constrained" models is challenging because standard uncertainty quantification (UQ) methods can violate these embedded physical conditions, require significant architectural changes, or incur high computational costs. This research introduces Structure-Preserving Epistemic Neural Networks (S-PENNs) to address this limitation. S-PENNs provide a general framework for UQ in scientific machine learning models that maintain hard architectural constraints. The method works by attaching lightweight "epinets" to the constrained components of a pre-trained model. This design ensures that every sampled realization remains physically admissible by construction. The framework was specifically instantiated for GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) dynamics, yielding thermodynamically consistent rollouts that preserve the first and second laws of thermodynamics. Furthermore, S-PENNs are combined with split conformal prediction for post-hoc calibration, providing prediction intervals with finite-sample marginal coverage guarantees. Validation on examples like a harmonic oscillator, a chemical motor, and a viscoplastic model demonstrated that S-PENNs produce thermodynamically consistent stochastic realizations and well-calibrated prediction intervals. Crucially, they achieve this while reducing computational costs by one to three orders of magnitude compared to deep ensembles, making them a highly efficient solution for UQ in physics-informed AI.