scientific machine learning engineers

Abstract: We ask whether physically meaningful correspondences discovered inside a trained scientific model remain meaningful outside the conditions under which they were discovered. We introduce GIIT (Graph-based Inspection and Intervention Tool), which represents governing physics as a computational physics dependency graph, maps graph nodes to internal network components via sensitivity- and trend-based discovery, and tests the resulting mapping under targeted intervention. Evaluating on temporal extrapolation out-of-distribution (OOD) regimes of increasing severity, we find that temporal extrapolation is associated with layer-wise correspondence drift and reduced functional correspondence. Specifically, while the model achieves low physics residual in-distribution (1.915 x 10-4 on full ID and 1.01 x 10-4 on an ID sub-window t in [0.5, 0.8]), the functional mapping changes even before leaving the training domain, and the shift continues as the evaluation window extends beyond the training domain: residual error increases from 2.81 x 10-2 on the boundary-crossing window (t in [0.5, 1.5]) to 3.30 x 10-1 on severe OOD (t in [1, 2]), accompanied by a systematic leftward shift of internal layer mappings, where the average winner layer index drops from 5.71 (ID) to 3.00 (ID sub-window) down to 2.29 (severe OOD). Only 2 of 7 physical nodes maintain stable layer assignments across the boundary-crossing window, and only 1 of 7 under severe extrapolation, revealing potential internal functional correspondence changes before severe degradation in conventional output metrics. Additional results for linear oscillatory systems are further detailed in the appendix.

Abstract: The Brinkman penalisation method embeds boundary-value problems on complex domains into a simple computational box by modeling the solid region as a strongly dissipative medium, avoiding body-fitted mesh generation. We show that multi-symplectic Hamiltonian PDEs regularised by Brinkman-type penalisation retain a multi-conformal symplectic structure under a compatibility condition linking the symplectic matrix and the penalisation projection. This yields an exact local conservation law, under which the multi-symplectic two-form is conserved in the fluid region and decays exponentially inside the solid. The linear wave equation with Brinkman friction and Maxwell's equations with artificial Ohmic conductivity satisfy this condition, with explicit modified Hamiltonian densities. Building on this, we propose (i) structure-preserving numerical integrators via Strang splitting that satisfy a discrete conformal conservation law, and (ii) conformal symplectic neural operators that interleave exact dissipative flows with learnable multi-symplectic evolution operators, allowing geometry-dependent operator learning. Numerical experiments on wave and electromagnetic scattering demonstrate that our methods reproduce correct local energy budgets and avoid unphysical energy drift, providing a principled framework for physics-consistent scientific machine learning on complex domains.

Abstract: We address the challenge of scalable uncertainty quantification in large-scale scientific applications, where complex state-of-the-art machine learning methods are often computationally infeasible. Our primary contribution is a simple yet effective empirical Bayes method for automatically tuning the hyperparameters of a flexible, heteroscedastic Spectral-normalized Neural Gaussian Process. This approach retains the expressiveness and uncertainty-awareness of semi-Bayesian neural models while significantly reducing the computational burden by integrating hyperparameter learning directly into the training loop. We demonstrate the practical impact of our method on the task of estimating the dynamic aperture in circular particle accelerators, a fundamental problem in high-energy physics colliders and storage rings, using simulation data from the case of the Large Hadron Collider at CERN. Traditional approaches to DA estimation require extensive particle-tracking simulations, which are prohibitively time-consuming and resource-intensive. Our results show that the proposed method achieves competitive predictive performance and well-calibrated uncertainty estimates at much lower computational cost than state-of-the-art approaches. We stress that, beyond this application, the proposed empirical Bayes framework offers a general solution for training heteroscedastic neural models in situations where manual hyperparameter tuning is impractical. Accordingly, we anticipate that this framework can be applied to other domains that encounter comparable computational limitations.