提升神经算子对微分方程正反问题的精度与鲁棒性
Sensitivity-Constrained Fourier Neural Operators for Forward and Inverse Problems in Parametric Differential Equations
- 引入敏感度约束正则化，增强对参数变化的响应建模能力
- 在82维参数空间下仍保持高精度，数据与训练量需求显著降低
- 适用于各类微分方程，适合需参数反演与快速求解场景
参数化微分方程 du/dt = f(u, x, t, p) 在科学与工程中具有基础地位。尽管深度学习框架如傅里叶神经算子（FNO）能高效近似解，但在反问题、敏感度估计（du/dp）和概念漂移方面表现不佳。本文提出基于敏感度的正则化策略——敏感度约束傅里叶神经算子（SC-FNO），显著提升解路径预测精度，优于标准FNO及物理信息正则化FNO。SC-FNO在参数反演任务中表现优异，可扩展至82维参数空间，大幅降低数据与训练需求。训练时间仅增加30%至130%每轮，且在多种微分方程与神经算子间具有良好泛化性。代码与部分实验已公开于 https://github.com/AMBehroozi/SC_Neural_Operators。
原文摘要 · Abstract (English)
Parametric differential equations of the form du/dt = f(u, x, t, p) are fundamental in science and engineering. While deep learning frameworks such as the Fourier Neural Operator (FNO) can efficiently approximate solutions, they struggle with inverse problems, sensitivity estimation (du/dp), and concept drift. We address these limitations by introducing a sensitivity-based regularization strategy, called Sensitivity-Constrained Fourier Neural Operators (SC-FNO). SC-FNO achieves high accuracy in predicting solution paths and consistently outperforms standard FNO and FNO with physics-informed regularization. It improves performance in parameter inversion tasks, scales to high-dimensional parameter spaces (tested with up to 82 parameters), and reduces both data and training requirements. These gains are achieved with a modest increase in training time (30% to 130% per epoch) and generalize across various types of differential equations and neural operators. Code and selected experiments are available at: https://github.com/AMBehroozi/SC_Neural_Operators
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