Neural non-canonical Hamiltonian dynamics for long-time simulations
Clémentine Courtès, Emmanuel Franck, Michael Kraus, Laurent Navoret, Léopold Trémant
TL;DR
This work addresses learning dynamics of non-canonical Hamiltonian systems from data with the goal of reliable long-time simulations. It shows that preserving geometric structure through learning is insufficient by itself if the numerical scheme introduces gauge-dependent errors, and proposes two complementary strategies: vector-field learning with a gauge-aware regularization, and scheme learning that directly fits a discrete, structure-preserving flow via a Degenerate Variational Integrator. Through experiments on Lotka–Volterra, a massless charged particle, and a guiding-center model, the authors demonstrate how regularization mitigates instability in vector-field learning, while scheme-learning yields superior long-time accuracy when the training time-step and scheme are aligned. The results highlight the importance of harmonizing learned structure, invariants, and the chosen integrator to achieve stable, accurate long-time behavior in non-canonical Hamiltonian dynamics, with applications to plasma physics and related areas.
Abstract
This work focuses on learning non-canonical Hamiltonian dynamics from data, where long-term predictions require the preservation of structure both in the learned model and in numerical schemes. Previous research focused on either facet, respectively with a potential-based architecture and with degenerate variational integrators, but new issues arise when combining both. In experiments, the learnt model is sometimes numerically unstable due to the gauge dependency of the scheme, rendering long-time simulations impossible. In this paper, we identify this problem and propose two different training strategies to address it, either by directly learning the vector field or by learning a time-discrete dynamics through the scheme. Several numerical test cases assess the ability of the methods to learn complex physical dynamics, like the guiding center from gyrokinetic plasma physics.