@article{46469,
  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. }},
  author       = {{Offen, Christian and Ober-Blöbaum, Sina}},
  issn         = {{1054-1500}},
  journal      = {{Chaos}},
  number       = {{1}},
  publisher    = {{AIP Publishing}},
  title        = {{{Learning of discrete models of variational PDEs from data}}},
  doi          = {{10.1063/5.0172287}},
  volume       = {{34}},
  year         = {{2024}},
}

@article{37654,
  abstract     = {{Recently, Hamiltonian neural networks (HNN) have been introduced to incorporate prior physical knowledge when
learning the dynamical equations of Hamiltonian systems. Hereby, the symplectic system structure is preserved despite
the data-driven modeling approach. However, preserving symmetries requires additional attention. In this research, we
enhance the HNN with a Lie algebra framework to detect and embed symmetries in the neural network. This approach
allows to simultaneously learn the symmetry group action and the total energy of the system. As illustrating examples,
a pendulum on a cart and a two-body problem from astrodynamics are considered.}},
  author       = {{Dierkes, Eva and Offen, Christian and Ober-Blöbaum, Sina and Flaßkamp, Kathrin}},
  issn         = {{1054-1500}},
  journal      = {{Chaos}},
  number       = {{6}},
  publisher    = {{AIP Publishing}},
  title        = {{{Hamiltonian Neural Networks with Automatic Symmetry Detection}}},
  doi          = {{10.1063/5.0142969}},
  volume       = {{33}},
  year         = {{2023}},
}

@article{16552,
  author       = {{Dellnitz, Michael and Hohmann, Andreas and Junge, Oliver and Rumpf, Martin}},
  issn         = {{1054-1500}},
  journal      = {{Chaos: An Interdisciplinary Journal of Nonlinear Science}},
  pages        = {{221--228}},
  title        = {{{Exploring invariant sets and invariant measures}}},
  doi          = {{10.1063/1.166223}},
  year         = {{1997}},
}