Locally-symplectic neural networks for learning volume-preserving dynamics
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We propose locally-symplectic neural networks LocSympNets for learning volume-preserving dynamics. The construction of LocSympNets stems from the theorem of local Hamiltonian description of the vector field of a volume-preserving dynamical system and the splitting methods based on symplectic integrators. Modified gradient modules of recently proposed symplecticity-preserving neural networks SympNets are used to construct locally-symplectic modules, which composition results in volume-preserving neural networks. LocSympNets are studied numerically considering linear and nonlinear dynamics, i.e., semi-discretized advection equation and Euler equations of the motion of a free rigid body, respectively. LocSympNets are able to learn linear and nonlinear dynamics to high degree of accuracy. When learning a single trajectory of the rigid body dynamics LocSympNets are able to learn both invariants of the system with absolute relative errors below 1 long-time predictions and produce qualitatively good short-time predictions, when the learning of the whole system from randomly sampled data is considered.
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