Symplectic Neural Operators for Learning Infinite Dimensional Hamiltonian Systems

2026-05-15

AI Summary

Modeling and simulating infinite-dimensional Hamiltonian systems poses significant challenges for conventional data-driven approaches in terms of computational efficiency and structure preservation. This work proposes the Symplectic Neural Operator, which, for the first time, embeds symplectic structure into the neural operator framework. The method is rigorously proven to preserve the intrinsic symplectic geometric structure of Hamiltonian partial differential equations, and a theoretical foundation is established that simultaneously guarantees structural fidelity and approximation accuracy, ensuring long-term stability. Experimental results on canonical Hamiltonian PDEs demonstrate that the proposed approach substantially outperforms non-structure-preserving methods, exhibiting superior energy conservation and remarkable stability in long-time 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.

Problem

• Hamiltonian systems
• infinite-dimensional
• symplectic structure
• neural operators
• long-term stability

Innovation

• Symplectic Neural Operator
• Hamiltonian PDEs
• structure-preserving learning
• infinite-dimensional systems
• long-term stability