Enforcing exact physics in scientific machine learning: A data-driven exterior calculus on graphs
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Trask, Nathaniel↗
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Scalable algorithms for physics-informed neural and graph networks
Physics-informed machine learning (PIML) has emerged as a promising new approach for simulating complex physical and biological systems that are governed by complex multiscale processes for which some data are also available. In some instances, the objective is to discover part of the hidden physics from the available data, and PIML has been shown to be particularly effective for such problems for which conventional methods may fail. Unlike commercial machine learning where training of deep neural networks requires big data, in PIML big data are not available. Instead, we can train such networks from additional information obtained by employing the physical laws and evaluating them at random points in the space–time domain. Such PIML integrates multimodality and multifidelity data with mathematical models, and implements them using neural networks or graph networks. Here, we review some of the prevailing trends in embedding physics into machine learning, using physics-informed neural networks (PINNs) based primarily on feed-forward neural networks and automatic differentiation. For more complex systems or systems of systems and unstructured data, graph neural networks (GNNs) present some distinct advantages, and here we review how physics-informed learning can be accomplished with GNNs based on graph exterior calculus to construct differential operators; we refer to these architectures as physics-informed graph networks (PIGNs). We present representative examples for both forward and inverse problems and discuss what advances are needed to scale up PINNs, PIGNs and more broadly GNNs for large-scale engineering problems.
42 ENGINEERING↗
QPatLib: A community repository of measurement-based quantum computing patterns
QPatLib is a community repository of measurement‑based quantum computing (MBQC) patterns and associated metadata, intended to support reuse, benchmarking, and reproducible comparison of MBQC constructions across platforms and compilers. Patterns in QPatLib are expressed in a standardized, human‑readable measurement calculus representation. Details can be found here: https://doi.org/10.48550/arXiv.2605.12502
graph states↗
Matching Curved Lattices to Anisotropic Tangent Planes
Radial quantization would be the ideal formalism for studying strongly-coupled near-conformal quantum field theories but it requires the ability to perform lattice calculations on static, curved manifolds, specifically a very long cylinder whose cross section is a sphere. Smoothly discretizing the surface of a sphere requires a graph with unequal edge lengths. The geometry of such graphs is well understood since 1961 using Regge Calculus. But, lattice quantum field theories are defined in terms of couplings which appear in the action rather than edge lengths and so the relationship between couplings and lengths must be determined dynamically. A simple example is computing the ratio of spatial to temporal lattice spacings in anisotropic lattice QCD. I will discuss our conjecture that computing anisotropic lattice spacing ratios on affine transformations of regular flat lattices is sufficient to determine coupling assignments on curved lattices.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Discovery of Probabilistic Dirichlet-to-Neumann Maps on Graphs
Dirichlet-to-Neumann maps enable the coupling of multiphysics simulations across computational subdomains by ensuring continuity of state variables and fluxes at artificial interfaces. We present a novel method for learning Dirichlet-to-Neumann maps on graphs using Gaussian processes, specifically for problems where the data obey a conservation law arising from an underlying partial differential equation. Our approach combines discrete exterior calculus and nonlinear optimal recovery to infer relationships between vertex and edge values. This framework yields data-driven predictions with uncertainty quantification across the entire graph, even when observations are limited to a subset of vertices and edges. By minimizing the reproducing kernel Hilbert space norm while penalizing kernel complexity through maximum likelihood estimation, our method ensures that the resulting surrogate strictly enforces conservation laws without overfitting. We demonstrate our method on two representative applications: subsurface flow in fracture networks and arterial blood flow. Finally, the results demonstrate that the method maintains high accuracy and well-calibrated uncertainty estimates even under severe data scarcity, highlighting its potential for scientific applications where limited data and reliable uncertainty quantification are critical.
Dirichlet-to-Neumann map↗
Data-driven Whitney forms for structure-preserving control volume analysis
Control volume analysis models physics via the exchange of generalized fluxes between subdomains. Here, we introduce a scientific machine learning framework adopting a partition of unity architecture to identify physically-relevant control volumes, with generalized fluxes between subdomains encoded via Whitney forms. The approach provides a differentiable parameterization of geometry which may be trained in an end-to-end fashion to extract reduced models from full field data while exactly preserving physics. The architecture admits a data-driven finite element exterior calculus allowing discovery of mixed finite element spaces with closed form quadrature rules. An equivalence between Whitney forms and graph networks reveals that the geometric problem of control volume learning is equivalent to an unsupervised graph discovery problem. The framework is developed for manifolds in arbitrary dimension, with examples provided for H(div) problems in $\mathbb{R}$ establishing convergence and structure preservation properties. Finally, we consider a lithium-ion battery problem where we discover a reduced finite element space encoding transport pathways from high-fidelity microstructure resolved simulations. The approach reduces the 5.89M finite element simulation to 136 elements while reproducing pressure to under 0.1% error and preserving conservation.
97 MATHEMATICS AND COMPUTING↗
Theoretical study of corrugated plates: Shear stiffness of a trapezoidally corrugated plate with discrete attachments to a rigid flange at the ends of the corrugations
Analysis and numerical results are presented for the elastic shear stiffness of a corrugated shear web with a certain type of discrete attachments at the ends of the trough lines of the corrugations, namely point attachments to a rigid flange which interferes with the deformations of the end cross sections by preventing downward movement but permitting upward (lifting off) movement. The analysis is based on certain assumed modes of deformation of the cross sections in conjunction with the method of minimum total potential energy and the calculus of variations in order to obtain equations for the manner in which the assumed modes of deformation vary along the length of the corrugation. The numerical results are restricted to the case of equal-width crests and troughs but otherwise apply to a wide variety of geometries. They are in the form of graphs which give the overall shear stiffness as a fraction of the overall shear stiffness that could be obtained by having continuous attachment at the ends of the corrugations.
Hsiao, C.↗
A Geometrical Approach to Bell's Theorem
Bell's theorem can be proved through simple geometrical reasoning, without the need for the Psi function, probability distributions, or calculus. The proof is based on N. David Mermin's explication of the Einstein-Podolsky-Rosen-Bohm experiment, which involves Stern-Gerlach detectors which flash red or green lights when detecting spin-up or spin-down. The statistics of local hidden variable theories for this experiment can be arranged in colored strips from which simple inequalities can be deduced. These inequalities lead to a demonstration of Bell's theorem. Moreover, all local hidden variable theories can be graphed in such a way as to enclose their statistics in a pyramid, with the quantum-mechanical result lying a finite distance beneath the base of the pyramid.
Rubincam, David Parry↗