用几何方法让神经网络高效模拟高维物理系统演化。
Learning Hamiltonian Dynamics at Scale: A Differential-Geometric Approach
- 构建可保持哈密顿结构的低维子流形，实现高效降维。
- 在复杂高维系统上实现能量守恒、长期稳定预测。
- 适合需要物理一致性与可扩展性的动力学建模任务。
将物理直觉嵌入网络架构，有助于学习满足能量守恒等基本规律的动力学模型，从而生成物理上合理的预测。然而，将此类模型扩展到高维动力系统仍面临重大挑战。本文提出一种新型物理启发式神经网络——约化阶哈密顿神经网络（RO-HNN），结合哈密顿力学的守恒律与模型降维的可扩展性。RO-HNN基于两个核心组件：一种新型几何约束的辛自编码器，用于学习低维、结构保持的辛子流形；以及一个定义在子流形上的几何哈密顿神经网络，用于建模动态过程。实验表明，RO-HNN能对复杂高维动力系统提供物理一致、稳定且可泛化的预测，有效拓展了哈密顿神经网络在高维物理系统中的应用范围。
原文摘要 · Abstract (English)
Embedding physical intuition into network architectures allows the learning of dynamics that enforce fundamental properties, such as energy conservation laws, thereby leading to physically-plausible predictions. Yet, scaling these models to high-dimensional dynamical systems remains a significant challenge. This paper introduces Reduced-order Hamiltonian Neural Network (RO-HNN), a novel physics-inspired neural network that combines the conservation laws of Hamiltonian mechanics with the scalability of model order reduction. RO-HNN is built on two core components: a novel geometrically-constrained symplectic autoencoder that learns a low-dimensional, structure-preserving symplectic submanifold, and a geometric Hamiltonian neural network that models the dynamics on the submanifold. Our experiments demonstrate that RO-HNN provides physically-consistent, stable, and generalizable predictions of complex high-dimensional dynamics, thereby effectively extending the scope of Hamiltonian neural networks to high-dimensional physical systems.
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