Abstract
The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density \(L_d\) that is modelled as a neural network. Careful regularisation of the loss function for training \(L_d\) is necessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of the discrete Euler–Lagrange equations. Secondly, we develop a method to find solutions to machine learned discrete field theories which constitute travelling waves of the underlying continuous PDE.
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Acknowledgements
C. Offen acknowledges the Ministerium für Kultur und Wissenschaft des Landes Nordrhein-Westfalen and computing time provided by the Paderborn Center for Parallel Computing (PC2).
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Offen, C., Ober-Blöbaum, S. (2023). Learning Discrete Lagrangians for Variational PDEs from Data and Detection of Travelling Waves. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14071. Springer, Cham. https://doi.org/10.1007/978-3-031-38271-0_57
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