用神经网络设计可保持能量守恒的长期物理模拟方法。
Hamiltonian Matching for Symplectic Neural Integrators
- 构建由时变哈密顿函数流映射组成的神经架构，保证相空间体积不变。
- 训练时通过匹配底层哈密顿量实现能量误差极低（10^-4量级）的长期演化。
- 适合需要长时间高精度模拟的物理系统，如天体力学和量子系统。
哈密顿运动方程是天体物理、量子力学、粒子物理和气候科学等多个领域的重要基础。传统数值求解器在处理多尺度时空系统时，累积误差会显著降低精度。为解决长时程演化的挑战，本文提出 SympFlow——一种基于神经网络的辛积分器，其结构由一系列参数化时变哈密顿函数的精确流映射构成。该架构支持反向误差分析：可识别出架构背后的隐含哈密顿量，并据此构建哈密顿量匹配目标函数用于训练。数值实验表明，SympFlow 在长期演化中表现出与时间步进辛积分器相当的能量守恒特性，能量偏差维持在 10^-4 量级，具备良好的定性稳定性。
原文摘要 · Abstract (English)
Hamilton's equations of motion form a fundamental framework in various branches of physics, including astronomy, quantum mechanics, particle physics, and climate science. Classical numerical solvers are typically employed to compute the time evolution of these systems. However, when the system spans multiple spatial and temporal scales numerical errors can accumulate, leading to reduced accuracy. To address the challenges of evolving such systems over long timescales, we propose SympFlow, a novel neural network-based symplectic integrator, which is the composition of a sequence of exact flow maps of parametrised time-dependent Hamiltonian functions. This architecture allows for a backward error analysis: we can identify an underlying Hamiltonian function of the architecture and use it to define a Hamiltonian matching objective function, which we use for training. In numerical experiments, we show that SympFlow exhibits promising results, with qualitative energy conservation behaviour similar to that of time-stepping symplectic integrators.
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