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.

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