Symplectic Neural Operators for Learning Infinite Dimensional Hamiltonian Systems

TL;DR

This paper introduces Symplectic Neural Operators, a neural architecture that preserves the symplectic structure of Hamiltonian PDEs, leading to improved stability and energy behavior in simulations.

Contribution

The paper proposes a novel neural operator architecture that maintains the symplectic structure of Hamiltonian systems, with theoretical analysis and empirical validation.

Findings

Symplectic Neural Operators preserve energy better than non-structure-preserving models.

Theoretical stability results are validated through numerical experiments.

SNOs demonstrate improved long-term stability in Hamiltonian PDE simulations.

Abstract

The modeling and simulation of infinite-dimensional Hamiltonian systems are central problems in mathematical physics and engineering, however they pose significant computational and structural challenges for standard data-driven architectures. In this work, we introduce the Symplectic Neural Operator, a neural operator architecture designed to preserve the symplectic structure intrinsic to Hamiltonian PDEs. We provide a theoretical characterization of their symplecticity and establish a rigorous long-term stability result based on the combination of symplectic structure preservation and learning accuracy. Numerical experiments on canonical Hamiltonian PDEs corroborate this theoretical result and show that SNOs exhibit improved energy behavior compared with non-structure-preserving neural operators.