用傅里叶神经算子加速变分数据同化中的海森逆计算
Meta-Learning Fourier Neural Operators for Hessian Inversion and Enhanced Variational Data Assimilation
- 用FNO元学习近似海森逆算子，为共轭梯度法提供高效初值
- 在线性平流方程上使相对误差降低62%，迭代次数减少17%
- 特别适合病态条件下的复杂数据同化问题，提升稳定性
数据同化（DA）对提升偏微分方程（如数值天气预报）的求解精度至关重要，通过观测数据优化初始条件。变分DA广泛用于海洋与大气预报，但涉及海森信息时计算成本高昂。本文提出一种元学习框架，利用傅里叶神经算子（FNO）逼近一族DA问题中的海森逆算子，从而为共轭梯度（CG）方法提供有效初始化。在线性平流方程上的数值实验表明，所提出的FNO-CG方法相较标准CG平均相对误差降低62%，迭代次数减少17%。该优势在病态场景中尤为显著，凸显FNO-CG在挑战性DA问题中的鲁棒性与高效性。
原文摘要 · Abstract (English)
Data assimilation (DA) is crucial for enhancing solutions to partial differential equations (PDEs), such as those in numerical weather prediction, by optimizing initial conditions using observational data. Variational DA methods are widely used in oceanic and atmospheric forecasting, but become computationally expensive, especially when Hessian information is involved. To address this challenge, we propose a meta-learning framework that employs the Fourier Neural Operator (FNO) to approximate the inverse Hessian operator across a family of DA problems, thereby providing an effective initialization for the conjugate gradient (CG) method. Numerical experiments on a linear advection equation demonstrate that the resulting FNO-CG approach reduces the average relative error by $62\%$ and the number of iterations by $17\%$ compared to the standard CG. These improvements are most pronounced in ill-conditioned scenarios, highlighting the robustness and efficiency of FNO-CG for challenging DA problems.
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