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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.22949 |
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| _version_ | 1866912609646673920 |
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| author | Moazzami, Hamidreza Jamali, Asma Kevlahan, Nicholas Vargas-Hernández, Rodrigo A. |
| author_facet | Moazzami, Hamidreza Jamali, Asma Kevlahan, Nicholas Vargas-Hernández, Rodrigo A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_22949 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Meta-Learning Fourier Neural Operators for Hessian Inversion and Enhanced Variational Data Assimilation Moazzami, Hamidreza Jamali, Asma Kevlahan, Nicholas Vargas-Hernández, Rodrigo A. Machine Learning 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. |
| title | Meta-Learning Fourier Neural Operators for Hessian Inversion and Enhanced Variational Data Assimilation |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2509.22949 |