提出可保持哈密顿系统结构的神经算子，长期模拟更稳定。
Symplectic Neural Operators for Learning Infinite Dimensional Hamiltonian Systems
- 设计保持辛结构的神经算子，适配无限维哈密顿方程。
- 理论证明其长期能量误差可控，数值实验验证稳定性提升。
- 适合需长期精确模拟的物理系统建模，如流体、场论。
无限维哈密顿系统的建模与仿真在数学物理和工程中至关重要，但对传统数据驱动架构带来显著计算与结构挑战。本文提出辛神经算子（Symplectic Neural Operator, SNO），一种旨在保持哈密顿偏微分方程内在辛结构的神经算子架构。我们提供了其辛性形式的理论刻画，并基于辛结构保持与学习精度的结合，建立了严格的长期稳定性结果。在典型哈密顿偏微分方程上的数值实验验证了该理论结果，表明SNO相较于不保持结构的神经算子展现出更优的能量行为。
原文摘要 · Abstract (English)
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.
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