Learning of discrete models of variational PDEs from data
arXiv:2308.05082 · doi:10.1063/5.0172287
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.
References in corpus (21)
- DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
- PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network
- A Comprehensive Physics-Informed Machine Learning Framework for Predictive Turbulence Modeling
- Physics-Informed Machine Learning Approach for Augmenting Turbulence Models: A Comprehensive Framework
- Variational methods, multisymplectic geometry and continuum mechanics
- On Learning Hamiltonian Systems from Data
- Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics
- Uniformly Accurate Machine Learning Based Hydrodynamic Models for Kinetic Equations
- Learning dynamical systems from data: a simple cross-validation perspective
- Degenerate Variational Integrators for Magnetic Field Line Flow and Guiding Center Trajectories
- On singular Lagrangian underlying the Schrödinger equation
- Variational Learning of Euler-Lagrange Dynamics from Data
- Symmetric Criticality in Classical Field Theory
- Symplectic model reduction methods for the Vlasov equation
- Projected Variational Integrators for Degenerate Lagrangian Systems
- Preserving Lagrangian structure in data-driven reduced-order modeling of large-scale dynamical systems
- Performance of the low-rank tensor-train SVD (TT-SVD) for large dense tensors on modern multi-core CPUs
- Hamiltonian Neural Networks with Automatic Symmetry Detection
- Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery
- Backward error analysis for variational discretisations of partial differential equations
- Backward error analysis for conjugate symplectic methods