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Approximation of Nearly-Periodic Symplectic Maps via Structure-Preserving Neural Networks
Valentin Duruisseaux
UCSD
Abstract:
A parametrized continuous-time dynamical system with parameter p is nearly-periodic if all its trajectories are periodic with nowhere-vanishing angular frequency as the value of p approaches 0. Nearly-periodic maps are discrete-time analogues of nearly-periodic systems, defined as parameter-dependent diffeomorphisms that limit to rotations along a circle action, and they admit formal U(1) symmetries to all orders when the limiting rotation is non-resonant. For Hamiltonian nearly-periodic maps on exact presymplectic manifolds, the formal U(1) symmetry gives rise to a discrete-time adiabatic invariant. In this talk, we will discuss a novel structure-preserving neural network architecture that we have recently proposed to approximate nearly-periodic symplectic maps. By design of this physics-informed deep learning architecture, which we call symplectic gyroceptron, near-periodicity and symplecticity are strongly enforced in the resulting surrogate map, which is also guaranteed to give rise to a discrete-time adiabatic invariant and long-time stability. This new structure-preserving neural network provides a promising architecture for surrogate modeling of non-dissipative dynamical systems that automatically steps over short timescales without introducing spurious instabilities.
Tuesday, November 1, 2022
11:00AM AP&M 2402 and Zoom ID 986 1678 1113
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Randolph E. Bank
Philip E. Gill
Michael Holst
Administrative Contact:
[ Click to Send Email ]
Valentin Duruisseaux
UCSD
Abstract:
A parametrized continuous-time dynamical system with parameter p is nearly-periodic if all its trajectories are periodic with nowhere-vanishing angular frequency as the value of p approaches 0. Nearly-periodic maps are discrete-time analogues of nearly-periodic systems, defined as parameter-dependent diffeomorphisms that limit to rotations along a circle action, and they admit formal U(1) symmetries to all orders when the limiting rotation is non-resonant. For Hamiltonian nearly-periodic maps on exact presymplectic manifolds, the formal U(1) symmetry gives rise to a discrete-time adiabatic invariant. In this talk, we will discuss a novel structure-preserving neural network architecture that we have recently proposed to approximate nearly-periodic symplectic maps. By design of this physics-informed deep learning architecture, which we call symplectic gyroceptron, near-periodicity and symplecticity are strongly enforced in the resulting surrogate map, which is also guaranteed to give rise to a discrete-time adiabatic invariant and long-time stability. This new structure-preserving neural network provides a promising architecture for surrogate modeling of non-dissipative dynamical systems that automatically steps over short timescales without introducing spurious instabilities.
Tuesday, November 1, 2022
11:00AM AP&M 2402 and Zoom ID 986 1678 1113