用结构保持方法预测耗散哈密顿系统，精度远超传统模型。
CoSynFlow: Conformal Symplectic Neural Flows for Cross-System Prediction of Dissipative Hamiltonian Dynamics
- 构建符合耗散系统几何特性的神经流模型，显式保留共形辛结构。
- 长期预测误差最低，结构误差控制在机器精度水平。
- 单个模型可泛化到未见系统，适合需物理一致性建模的场景。
学习微分方程解算子是科学机器学习的核心问题。然而，许多神经算子方法仅优化预测精度，未显式保留动力学的几何结构。针对保守哈密顿系统，如SympNets和辛神经流等结构保持模型通过保辛形式解决此问题。但在具有共形辛结构的耗散哈密顿系统中，辛形式随耗散因子演化。本文提出CoSynFlow，一种用于学习耗散哈密顿动力学连续时间解映射的共形辛神经流。CoSynFlow通过组合保辛剪切变换与显式共形缩放，构造性地保持共形辛结构。通过条件化于有限维哈密顿描述符和耗散参数，单一训练模型无需重训即可预测未见系统的解映射。CoSynFlow将结构误差保持在机器精度，实现最低的长期预测误差，并支持物理信息训练。
原文摘要 · Abstract (English)
Learning solution operators for differential equations is a central problem in scientific machine learning. However, many neural operator methods optimize prediction accuracy without explicitly enforcing the geometric structure of the dynamics. Structure-preserving models such as SympNets and Symplectic Neural Flows address this issue for conservative Hamiltonian systems by preserving the symplectic form. In dissipative Hamiltonian systems with conformal symplectic structure, however, the symplectic form evolves according to a conformal factor determined by the dissipation. We propose CoSynFlow, a conformal symplectic neural flow for learning continuous-time solution maps of dissipative Hamiltonian dynamics. CoSynFlow composes symplectic shear maps with explicit conformal scaling, preserving the conformal symplectic structure by construction. By conditioning it on a finite-dimensional Hamiltonian descriptor and the dissipation parameter, a single trained model predicts solution maps for unseen systems without retraining. CoSynFlow keeps the structure error at machine precision, attains the lowest long-horizon error, and admits physics-informed training.
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