发表机构
Laboratory of Mathematics and Complex Systems, Ministry of Education and School of Mathematical Sciences, Beijing Normal University（数学与复杂系统教育部重点实验室，北京师范大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究将深度学习与保结构指数时间差分方法结合，引入轨迹推断自由能，实现从轨迹数据的保结构自由能学习，可准确恢复能量并进行可靠长时间预测。
AI 中文摘要
当控制方程和自由能已知时，梯度流的保结构数值方法已得到广泛发展，但从观测数据准确预测动力学同时保留内在物理性质仍具挑战性。尽管傅里叶神经算子（FNO）等神经算子提供了强大的数据驱动近似，但其预测本身无法保证物理结构的保留。为解决该挑战，我们将深度学习与保结构指数时间差分（ETD）方法相结合，未直接学习演化算子，而是引入轨迹推断自由能（TIFE），通过基于杜哈梅尔（Duhamel）的学习框架从观测轨迹重构未知能量密度。学习到的能量决定变分力和稳定参数，实现保结构数值演化。我们进一步将框架扩展至联合识别自由能与扩散系数，在适当假设下建立最大模原理、原始学习能量的无条件耗散及包含离散化与推断误差的误差估计。数值实验表明该方法可实现准确的能量恢复与可靠的长时间预测。
英文摘要
Structure-preserving numerical methods for gradient flows have been extensively developed when the governing equations and free energies are known. However, accurately predicting dynamics from observational data while preserving intrinsic physical properties remains challenging. Although neural operators, such as Fourier neural operators (FNOs), provide powerful data-driven approximations, their predictions do not inherently guarantee physical structure preservation. To address this challenge, we integrate deep learning with structure-preserving exponential time differencing (ETD) methods. Instead of directly learning the evolution operator, we introduce a *trajectory-inferred free energy* (TIFE), which reconstructs the unknown energy density from observed trajectories through a Duhamel-based learning framework. The learned energy determines the variational force and stabilization parameter, enabling structure-preserving numerical evolution. We further extend the framework to jointly identify the free energy and diffusion coefficient. Under suitable assumptions, we establish the maximum-bound principle, unconditional dissipation of the original learned energy, and error estimates incorporating discretization and inference errors. Numerical experiments demonstrate accurate energy recovery and reliable long-time predictions.
Comments32 pages, 11 figures
