Neural non-canonical Hamiltonian dynamics for long-time simulations

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. Keywords Hamiltonian dynamics, structure preserving integrator, symplectic integrators, neural models, geometric machine learning, guiding center Subject classification (MSC2020) 65P10, 68T07, 37M15 1 Introduction With the rise in machine learning research, the connection between neural networks and differential equations--particularly ordinary differential equations (ODEs)--has become an increasingly active area of study, explored through three main application domains. The first approach interprets certain neural networks, such as Residual Networks (ResNets), as discretizations of ODEs [CRBD18, DDT19, MPP + 20]. This perspective not only enables the design of novel architectures but also provides deeper insight into the internal dynamics of neural networks.