保结构傅里叶神经算子用于粗时间监督下的长时间Cahn-Hilliard动力学
Structure-preserving Fourier Neural Operators for Long-time Cahn-Hilliard Dynamics under Coarse Temporal Supervision
浏览论文内容
中文总结 AI 辅助
针对Cahn-Hilliard方程长时间粗化动力学预测误差累积问题，提出两阶段保结构傅里叶神经算子框架，通过边界一致与结构函数正则化修正后期统计，提升长时间预测精度。
中文摘要 AI 辅助
Cahn-Hilliard动力学的长时间精确预测，特别是在后期粗化阶段，由于需要在延长的物理时间上进行模拟，计算成本很高。人们开发了各种机器学习方法，通过高效的代理评估来加速模拟。虽然学习到的映射在单个学习区间内是准确的，但当该映射在长时间预测中被反复应用时，小的预测误差会累积，导致较大的数值误差。在本工作中，我们识别了Cahn-Hilliard方程粗化动力学的结构函数，并提出了一种两阶段保结构傅里叶神经算子（FNO）学习框架。第一阶段引入了一种具有软性、幅度依赖幅度惩罚的边界一致FNO（bcFNO）。第二阶段保留边界一致目标，并补充结构函数正则化，得到一种保结构FNO（spFNO），用于修正后期粗化统计。在不同空间分辨率和参考求解器时间步长上的数值实验表明，bcFNO通过抑制大幅度偏移稳定了长时间预测，同时不改变FNO的低在线成本。此外，结构函数项持续减少了归一化结构函数的差异，并在相关的后期粗化量中提供了进一步修正，包括$L^3(t)$增长趋势和标度坍缩。这些结果表明，我们在算子学习中多阶段实现物理原理的方法可以应用于其他具有统计标度的多尺度系统，例如功能化Cahn-Hilliard方程和湍流，前提是状态约束和统计观测量适应于主导动力学。
英文摘要
Accurate long-time prediction of Cahn-Hilliard dynamics, particularly in the late-stage coarsening regime, is computationally demanding because it requires simulations over extended physical time. Various machine learning approaches have been developed to speed up the simulations through efficient surrogate evaluations. While the learned map can be accurate over a single learned interval, small prediction error will accumulate when the map is repeatedly applied in a long-time prediction, leading to large numerical errors. In this work, we identify the structure function for the coarsening dynamics of the Cahn-Hilliard equation and propose a two-stage structure-preserving Fourier Neural Operator (FNO) learning framework. Stage~I introduces a bound-conforming FNO (bcFNO) with a soft, magnitude-dependent amplitude penalty. Stage~II retains the bound-conforming objective and supplements it with structure-function regularization, yielding a structure-preserving FNO (spFNO) that corrects late-stage coarsening statistics. Numerical experiments on the various spatial resolutions and reference-solver time steps suggest that the bcFNO stabilizes the long-time prediction by suppressing large amplitude excursions, without changing the low online cost of the FNO. Furthermore, the structure-function term consistently reduces discrepancies in the normalized structure function and provides further correction in associated late-stage coarsening quantities, including the $L^3(t)$ growth trend and the scaling collapse. These results indicate that our multi-stage implementation of physical principles in operator learning could be applied to other multiscale systems with statistical scaling, such as the functionalized Cahn--Hilliard equations and turbulence, provided that the state constraints and statistical observables are adapted to the governing dynamics.
发表机构
- Colorado State University（科罗拉多州立大学）
机构由 AI 辅助整理，请以论文原文为准。
