Meta-Learning Fourier Neural Operators for Hessian Inversion and Enhanced Variational Data Assimilation
This work addresses efficiency challenges in data assimilation for numerical weather prediction, though it is incremental as it builds on existing methods like FNO and CG.
The paper tackled the computational expense of Hessian inversion in variational data assimilation by proposing a meta-learning framework using Fourier Neural Operators to approximate the inverse Hessian, reducing average relative error by 62% and iterations by 17% in experiments on a linear advection equation.
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.