Thermodynamics-informed graph neural networks

Hernández, Quercus; Badías, Alberto; Chinesta, Francisco; Cueto, Elías

doi:10.48550/arxiv.2203.01874

“…We comment, however, that this notion of compatibility is appropriate for nonlinear elliptic problems. Other notions of compatibility are important for other problems; for example, geometric mechanics is important for dynamical systems while for flow problems it is necessary to consider Lie derivatives for advection operators (Marsden et al, 1998;Lee et al, 2021;Hernández et al, 2022).…”

Section: Graph Exterior Calculusmentioning

confidence: 99%

Scalable algorithms for physics-informed neural and graph networks

Shukla

¹

,

Xu

²

,

Trask

³

et al. 2022

DCE

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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.

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“…We comment, however, that this notion of compatibility is appropriate for non-linear elliptic problems. Other notions of compatibility are important for other problems; for example, geometric mechanics is important for dynamical systems while for flow problems it is necessary to consider Lie derivatives for advection operators (Hernández et al, 2022;Lee et al, 2021;Marsden et al, 1998).…”

Section: Graph Exterior Calculusmentioning

confidence: 99%

Scalable algorithms for physics-informed neural and graph networks

Shukla¹,

Xu²,

Trask³

et al. 2022

Preprint

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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 physics-informed machine learning 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. Impact StatementMany complex problems in computational engineering can be described by parametrized differential equations while boundary conditions or material properties may not be fully known, e.g., in thermal-fluid or solid mechanics systems. These ill-defined problems cannot be solved with standard numerical methods but if some sparse data are available, progress can be made by recasting them as large-scale minimization problems. Here, we review physics-informed neural networks (PINNs) and physics-informed graph networks (PIGNs) that integrate seamlessly data and mathematical physics models, even in partially understood or uncertain contexts. Using automatic differentiation in PINNs and external graph calculus in PIGNs, the physical laws are enforced by penalizing the residuals on random points in the space-time domain for PINNs and on the nodes of a graph in PIGNs. New multi-GPU algorithms are needed to scale up PINNs and PIGNs to realistic applications.

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“…This formulation gave rise to the so-called structure-preserving neural networks [116] and thermodynamics-informed neural networks [117,118,119]. These networks have been employed recently in the development of physcs perception with the help of computer vision techniques [120,121].…”

Section: Metriplectic Neural Networkmentioning

confidence: 99%

“…An approach has been developed in which both inductive biases for the thermodynamic structure of the problem and a graph structure in the network are used in conjunction [119]. This approach has demonstrated to be very convenient for the development of learned simulators trained from finite element data.…”

Section: Metriplectic Neural Networkmentioning

confidence: 99%

Thermodynamics of learning physical phenomena

Chinesta¹

2022

Preprint

Self Cite

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Thermodynamics could be seen as an expression of physics at a high epistemic level. As such, its potential as an inductive bias to help machine learning procedures attain accurate and credible predictions has been recently realized in many fields. We review how thermodynamics provides helpful insights in the learning process. At the same time, we study the influence of aspects such as the scale at which a given phenomenon is to be described, the choice of relevant variables for this description or the different techniques available for the learning process.

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Thermodynamics-informed graph neural networks

Cited by 6 publications

References 28 publications

Scalable algorithms for physics-informed neural and graph networks

Scalable algorithms for physics-informed neural and graph networks

Scalable algorithms for physics-informed neural and graph networks

Thermodynamics of learning physical phenomena

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