Enforcing exact physics in scientific machine learning: a data-driven exterior calculus on graphs

Abstract

As traditional machine learning tools are increasingly applied to science and engineering applications, physics-informed methods have emerged as effective tools for endowing inferences with properties essential for physical realizability. While promising, these methods generally enforce physics weakly via penalization. To enforce physics strongly, we turn to the exterior calculus framework underpinning combinatorial Hodge theory and physics-compatible discretization of partial differential equations (PDEs). Historically, these two fields have remained largely distinct, as graphs are strictly topological objects lacking the metric information fundamental to PDE discretization. We present an approach where this missing metric information may be learned from data, using graphs as coarse-grained mesh surrogates that inherit desirable conservation and exact sequence structure from the combinatorial Hodge theory. The resulting datadriven exterior calculus (DDEC) may be used to extract structure-preserving surrogate models with mathematical guarantees of well-posedness. The approach admits a PDE-constrained optimization training strategy which guarantees machine-learned models enforce physics to machine precision, even for poorly trained models or small data regimes. We provide analysis of the method for a class of models designed to reproduce nonlinear perturbations of elliptic problems and provide examples of learning H(div)/H(curl) systems representative of subsurface flows and electromagnetics.

Machine learning structure preserving brackets for forecasting irreversible processes

“…More recently, there has been efforts to add physical invariance to learned dynamics models, e.g., time-reversal symmetry [37]. Works pursuing related but distinct spatial-compatibility related to conservation structure other than geometric integration include: graph architectures with associated a data-driven graph exterior calculus [38], solving optimization problems with conservation constraint in latent space [39], and adding conservation constraints as a penalty in training loss [40]. The closest work to our approach is in [41], which proposed a time integrator that leverages the GENERIC (general equation for the nonequilibrium reversible-irreversible coupling) formalism to impose the structure, but enforces the degeneracy condition as penalty terms in the training loss objective.…”

Structure-preserving Sparse Identification of Nonlinear Dynamics for Data-driven Modeling

“…image/language processing tasks, engineering and science models pose strict requirements on physical quantities to guarantee properties such as conservation, thermodynamic consistency and well-posedness of resulting models (Baker et al 2019). These structural constraints translate to desirable mathematical properties for simulation, such as improved numerical stability and accuracy, particularly for chaotic systems (Lee, Trask, and Stinis 2021;Trask, Huang, and Hu 2020).…”

“…While so-called physics-informed ML (PIML) approaches seek to impose these properties by imposing soft physics constraints into the ML process, many applications require structure preservation to hold exactly; PIML requires empirical tuning of weighting parameters and physics properties hold only to within optimization error, which typically may be large (Wang, Teng, and Perdikaris 2020;Rohrhofer, Posch, and Geiger 2021). Structure-preserving machine learning has emerged as a means of designing architectures such that physics constraints hold exactly by construction (Lee, Trask, and Stinis 2021;Trask, Huang, and Hu 2020). By parameterizing relevant geometric or topological structures, researchers obtain more data-efficient hybrid physics/ML architectures with guaranteed mathematical properties.…”

Mesh-based graph convolutional neural networks for modeling materials with microstructure

“…Zhang et al [33], and Zhou et al [34]. Note that related kernel-based NN methods, such as the work of Trask et al [35,36] exists and have been shown to be effective in physical applications. Also, Bronstein et al [28] provides an insightful perspective on applications of deep learning to non-Euclidean data, graphs, and manifolds as well as the relevant mathematics.…”