A variational latent neural field framework that integrates Gaussian process-inspired surrogates is developed, enabling estimation of predictive confidence for both in-distribution and out-of-distribution parameter regimes and shows that conservation-structure preserving latent representations improve robustness to degraded training data while maintaining competitive predictive accuracy and uncertainty quantification capability.
Abstract
Reduced-order models, such as latent dynamics models, are becoming mainstream for accelerating simulations for parameterized physical systems governed by nonlinear conservation laws. However, most existing latent dynamics frameworks suffer from two important limitations: they do not provide uncertainty estimates for model predictions, and they do not guarantee adherence to the underlying conservation laws. While these challenges have been addressed separately in prior work, a unified framework that simultaneously provides uncertainty quantification and exact conservation-law preservation remains largely unexplored. In this work, we develop a variational latent neural field framework that integrates Gaussian process-inspired surrogates, enabling estimation of predictive confidence for both in-distribution and out-of-distribution parameter regimes. Three variants of the framework are considered: IRS-UQ, PI-IRS-UQ, and ECLEIRS-UQ, corresponding to unconstrained, physics-informed, and conservation-structure-preserving formulations, respectively. Exact conservation-structure preservation is achieved by embedding the solution dynamics within a conservation-law manifold through a space-time divergence-free representation of the solution-flux field. We demonstrate the applicability of the framework through three numerical experiments: 1) 1-D advection, 2) 2-D Euler and 3) 2-D shallow water equations in parameterized settings. Numerical experiments demonstrate that the proposed approach provides accurate predictions together with uncertainty estimates, while remaining robust to sparse and noisy training data. Comparisons between the proposed three approaches show that conservation-structure preserving latent representations improve robustness to degraded training data while maintaining competitive predictive accuracy and uncertainty quantification capability.
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.
Experiments show that FC-VPINN achieves approximately one-order-of-magnitude lower prediction errors than the traditional PINN and reduces memory usage to 40% of that required by the baseline, demonstrating improved accuracy and computational efficiency in multidimensional problems.
Wenjie Zhang, Yu-Bo Li, Wei-Dong Cui et al.· Chinese Physics B· 0 citations
A physics-aware neural-network-based latent-space framework for reduced-order forward modeling and variational parameter estimation and the robustness of the proposed framework is demonstrated, which generally reduces calibration error and variability compared with the standard surrogate models.
Qiyao Zhou, Xujia Zhu, Pierre Joli et al.· 0 citations
A regularized PINNs framework that incorporates Tikhonov regularization to solve terminal-state tracking optimal control constrained by parabolic partial differential equations is proposed, establishing a consistency result showing that PINNs minimizers nearly attain the continuous regularized objective under residual and quadrature approximation assumptions.
We address the simultaneous prediction of multiple high-dimensional physical fields governed by linear equality constraints, a setting that arises in many real-world applications in physics machine learning. Gaussian process (GP) regression is a widely used surrogate modeling approach due to its effectiveness in small-sample regimes and its ability to provide uncertainty quantification. However, applying GP models in this setting raises two major challenges: the high dimensionality of the discretized output fields and the enforcement of the physical constraint in predictions. For the latter, a common strategy consists in deducing one output from the others via the constraint relation. Through a benchmark, we show that this deductive approach is sensitive to the arbitrary choice of which output to deduce, affecting both predictive accuracy and uncertainty quantification. Consequently, there is a need for an approach that treats all fields symmetrically while strictly respecting the underlying physics. Motivated by these limitations, we propose a robust framework for jointly modeling constrained multi-field data. Our approach first leverages a specific PCA procedure for multi-field data, coined row-wise PCA, which has the interesting property of preserving the constraint in the latent space. Since standard PCA strategies for multi-field data do not preserve such constraints, we investigate theoretically the optimality of the row-wise choice. In a second step, we consider a linearly-constrained multi-output GP approach based on a specific kernel parametrization which is trained on the latent space of row-wise PCA. The proposed framework is validated on a population dynamics problem and on an industrial CFD application, which involves the prediction of Reynolds stress tensor components under the incompressibility constraint.
Mahamat Hamdan Nassouradine, C. Gauchy, Pierre-Emmanuel Angeli et al.· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
