Learning of discrete models of variational PDEs from data

C. Offen, S. Ober-Blöbaum, Chaos (n.d.).

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Journal Article | Accepted | English
Abstract
We show how to learn discrete field theories from observational data of fields on a space-time lattice. For this, we train a neural network model of a discrete Lagrangian density such that the discrete Euler--Lagrange equations are consistent with the given training data. We, thus, obtain a structure-preserving machine learning architecture. Lagrangian densities are not uniquely defined by the solutions of a field theory. We introduce a technique to derive regularisers for the training process which optimise numerical regularity of the discrete field theory. Minimisation of the regularisers guarantees that close to the training data the discrete field theory behaves robust and efficient when used in numerical simulations. Further, we show how to identify structurally simple solutions of the underlying continuous field theory such as travelling waves. This is possible even when travelling waves are not present in the training data. This is compared to data-driven model order reduction based approaches, which struggle to identify suitable latent spaces containing structurally simple solutions when these are not present in the training data. Ideas are demonstrated on examples based on the wave equation and the Schrödinger equation.
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Chaos
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Offen C, Ober-Blöbaum S. Learning of discrete models of variational PDEs from data. Chaos.
Offen, C., & Ober-Blöbaum, S. (n.d.). Learning of discrete models of variational PDEs from data. Chaos.
@article{Offen_Ober-Blöbaum, title={Learning of discrete models of variational PDEs from data}, journal={Chaos}, publisher={AIP Publishing}, author={Offen, Christian and Ober-Blöbaum, Sina} }
Offen, Christian, and Sina Ober-Blöbaum. “Learning of Discrete Models of Variational PDEs from Data.” Chaos, n.d.
C. Offen and S. Ober-Blöbaum, “Learning of discrete models of variational PDEs from data,” Chaos.
Offen, Christian, and Sina Ober-Blöbaum. “Learning of Discrete Models of Variational PDEs from Data.” Chaos, AIP Publishing.
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Learning of discrete models of variational PDEs from data
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Effective machine learning architectures for dynamical systems are in demand: data-driven models can be used to predict a system's behaviour when no analytic model is available or be employed as surrogates when analytical models are too expensive to simulate. The article presents a new technique to learn field theories from data in a structure preserving way. Central novelties are the consistent use of discrete variational theory in the design of the machine learning architectures and the systematic embedding of ideas from numerical integration theory. Our technique allows for the data-driven identification of travelling waves of the field theory even when these have not been in the training data set. On this task approaches in the literature fail.
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Learning of discrete models of variational PDEs from data (single column format)
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Single column format of article "Learning of discrete models of variational PDEs from data"
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