Structure-preserving uncertainty quantification for GENERIC dynamics

Scientific machine learning models often impose physical laws—such as the first and second laws of thermodynamics—as hard constraints built directly into the model architecture. While these structure-preserving models ensure physical admissibility, quantifying their predictive uncertainty remains challenging. Standard uncertainty quantification (UQ) methods like deep ensembles or Bayesian neural networks either break the hard constraints, require expensive architectural modifications, or demand prohibitive computational costs because generic weight perturbations do not respect the underlying geometric and thermodynamic invariant spaces.

This paper introduces Structure-Preserving Epistemic Neural Networks (S-PENNs), a general framework that injects epistemic uncertainty into constrained machine learning models without violating their structural properties. The authors instantiate this framework for nonlinear GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) dynamics, which decompose physical evolution into reversible Hamiltonian parts and irreversible generalized gradient flows.

Instead of training independent full models, S-PENNs treat a deterministic structure-preserving network (N-GENN) as a base network and attach separate, lightweight auxiliary networks called epinets to each thermodynamic building block. Each epinet is conditioned on stop-gradient features and driven by a shared epistemic index. Because these perturbed components are reassembled through the original structure-preserving operators—such as skew-symmetric Poisson matrices and partially input-convex neural networks—every sampled stochastic realization remains thermodynamically admissible by construction. Furthermore, the framework integrates split conformal prediction as a post-hoc calibration step to deliver prediction intervals with finite-sample marginal coverage guarantees.

The proposed S-PENN framework was evaluated on three numerical benchmarks: a harmonic oscillator coupled to a heat bath, an idealized chemical motor, and a one-dimensional viscoplastic model. Across all cases, S-PENNs successfully generated thermodynamically consistent stochastic rollouts and well-calibrated prediction intervals. Crucially, the method achieved these results while reducing computational training costs by one to three orders of magnitude compared to standard deep ensembles. This demonstrates that rigorous UQ can be efficiently combined with hard physical constraints, opening pathways for reliable predictive modeling in computational mechanics.