Locally-symplectic neural networks for learning volume-preserving dynamics
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Publication:6378004
DOI10.1016/J.JCP.2023.111911arXiv2109.09151MaRDI QIDQ6378004
Publication date: 19 September 2021
Abstract: We propose locally-symplectic neural networks LocSympNets for learning the flow of phase volume-preserving dynamics. The construction of LocSympNets stems from the theorem of the local Hamiltonian description of the divergence-free vector field and the splitting methods based on symplectic integrators. Symplectic gradient modules of the recently proposed symplecticity-preserving neural networks SympNets are used to construct invertible locally-symplectic modules. To further preserve properties of the flow of a dynamical system LocSympNets are extended to symmetric locally-symplectic neural networks SymLocSympNets, such that the inverse of SymLocSympNets is equal to the feed-forward propagation of SymLocSympNets with the negative time step, which is a general property of the flow of a dynamical system. LocSympNets and SymLocSympNets are studied numerically considering learning linear and nonlinear volume-preserving dynamics. We demonstrate learning of linear traveling wave solutions to the semi-discretized advection equation, periodic trajectories of the Euler equations of the motion of a free rigid body, and quasi-periodic solutions of the charged particle motion in an electromagnetic field. LocSympNets and SymLocSympNets can learn linear and nonlinear dynamics to a high degree of accuracy even when random noise is added to the training data. When learning a single trajectory of the rigid body dynamics locally-symplectic neural networks can learn both quadratic invariants of the system with absolute relative errors below 1%. In addition, SymLocSympNets produce qualitatively good long-time predictions, when the learning of the whole system from randomly sampled data is considered. LocSympNets and SymLocSympNets can produce accurate short-time predictions of quasi-periodic solutions, which is illustrated in the example of the charged particle motion in an electromagnetic field.
Artificial intelligence (68Txx) Numerical methods for ordinary differential equations (65Lxx) Hamiltonian and Lagrangian mechanics (70Hxx)
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