# Meta-Learning Fourier Neural Operators for Hessian Inversion and Enhanced Variational Data Assimilation

arXiv: https://arxiv.org/abs/2509.22949 · HTML: https://mlsanity.com/paper/2509.22949/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 25 references that are in the map.
- First posted on arXiv 26 Sep 2025.

## Builds on (most cited first, 8 of 8)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Meta-Learning in Neural Networks: A Survey](https://mlsanity.com/paper/2004.05439/) (2020)
- [Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport](https://mlsanity.com/paper/1912.02968/) (2019)
- [4D-Var using Hessian approximation and backpropagation applied to automatically-differentiable numerical and machine learning models](https://mlsanity.com/paper/2408.02767/) (2024)
- [Learning Semilinear Neural Operators : A Unified Recursive Framework For Prediction And Data Assimilation](https://mlsanity.com/paper/2402.15656/) (2024)
- [POD/DEIM Reduced-Order Strategies for Efficient Four Dimensional Variational Data Assimilation](https://mlsanity.com/paper/1402.5992/) (2014)
- [Efficient methods for computing observation impact in 4D-Var data assimilation](https://mlsanity.com/paper/1305.5796/) (2013)
- [A Numerical Study of Chaotic Dynamics of K-S Equation with FNOs](https://mlsanity.com/paper/2410.12280/) (2024)

## Abstract (arXiv)

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.