# Structure-preserving uncertainty quantification for GENERIC dynamics

arXiv: https://arxiv.org/abs/2608.12624 · HTML: https://mlsanity.com/paper/2608.12624/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 41 of its 81 references that are in the map.
- First posted on arXiv 12 Aug 2026.

## Builds on (most cited first, 30 of 41)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [A Survey of Uncertainty in Deep Neural Networks](https://mlsanity.com/paper/2107.03342/) (2021)
- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Accurate Uncertainties for Deep Learning Using Calibrated Regression](https://mlsanity.com/paper/1807.00263/) (2018)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users](https://mlsanity.com/paper/2007.06823/) (2020)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Bayesian Neural Networks: An Introduction and Survey](https://mlsanity.com/paper/2006.12024/) (2020)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Epistemic Neural Networks](https://mlsanity.com/paper/2107.08924/) (2021)
- [Evaluating and Calibrating Uncertainty Prediction in Regression Tasks](https://mlsanity.com/paper/1905.11659/) (2019)
- [Structure-preserving neural networks](https://mlsanity.com/paper/2004.04653/) (2020)
- [Machine learning structure preserving brackets for forecasting irreversible processes](https://mlsanity.com/paper/2106.12619/) (2021)
- [GFINNs: GENERIC Formalism Informed Neural Networks for Deterministic and Stochastic Dynamical Systems](https://mlsanity.com/paper/2109.00092/) (2021)
- [Learning Poisson systems and trajectories of autonomous systems via Poisson neural networks](https://mlsanity.com/paper/2012.03133/) (2020)
- [OnsagerNet: Learning Stable and Interpretable Dynamics using a Generalized Onsager Principle](https://mlsanity.com/paper/2009.02327/) (2020)
- [Structure preserving deep learning](https://mlsanity.com/paper/2006.03364/) (2020)
- [Approximate Thompson Sampling via Epistemic Neural Networks](https://mlsanity.com/paper/2302.09205/) (2023)
- [Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operators](https://mlsanity.com/paper/2311.11262/) (2023)
- [Bayesian-EUCLID: discovering hyperelastic material laws with uncertainties](https://mlsanity.com/paper/2203.07422/) (2022)
- [PI-VAE: Physics-Informed Variational Auto-Encoder for stochastic differential equations](https://mlsanity.com/paper/2203.11363/) (2022)
- [Efficiently Parameterized Neural Metriplectic Systems](https://mlsanity.com/paper/2405.16305/) (2024)

## Abstract (arXiv)

Structure-preserving machine learning embeds physical structure directly into model architectures, yet uncertainty quantification (UQ) for such hard-constrained models remains limited because standard UQ methods may violate the encoded admissibility conditions, require architectural modifications, or impose substantial computational costs. In this work, we propose Structure-Preserving Epistemic Neural Networks (S-PENNs), a general framework for UQ in scientific machine learning models with hard architectural constraints, and instantiate it for GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) dynamics. S-PENNs preserve the structural constraints of a pretrained model by attaching lightweight epinets to its constrained components, ensuring that every sampled realization remains physically admissible by construction. When applied to GENERIC dynamics, such a proposed framework yields thermodynamically consistent rollouts that preserve the first and second laws. Furthermore, we combine S-PENNs with split conformal prediction as a post-hoc calibration method to produce prediction intervals with finite-sample marginal coverage guarantees. We validate S-PENNs on three numerical examples: a harmonic oscillator coupled to a heat bath and an idealized chemical motor, both governed by ODEs, and a one-dimensional viscoplastic model governed by PDEs. Across all three examples, S-PENNs produce thermodynamically consistent stochastic realizations and well-calibrated prediction intervals while reducing the computational cost by about one to three orders of magnitude compared to deep ensembles. Although the present study focuses on GENERIC dynamics, S-PENNs can be extended more broadly to scientific machine learning models in computational mechanics with either hard or soft constraints.