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

• URL: http://arxiv.org/abs/2509.22949v1
• Date: Fri, 26 Sep 2025 21:30:31 GMT
• Title: Meta-Learning Fourier Neural Operators for Hessian Inversion and Enhanced Variational Data Assimilation
• Authors: Hamidreza Moazzami, Asma Jamali, Nicholas Kevlahan, Rodrigo A. Vargas-Hernández,
• Abstract summary: 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.<n> 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.
• Score: 0.6999740786886536
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

Related papers

• Multi-Dimensional Visual Data Recovery: Scale-Aware Tensor Modeling and Accelerated Randomized Computation [51.65236537605077]
  We propose a new type of network compression optimization technique, fully randomized tensor network compression (FCTN)<n>FCTN has significant advantages in correlation characterization and transpositional in algebra, and has notable achievements in multi-dimensional data processing and analysis.<n>We derive efficient algorithms with guarantees to solve the formulated models.
  arXiv  Detail & Related papers  (2026-02-13T14:56:37Z)
• DAISI: Data Assimilation with Inverse Sampling using Stochastic Interpolants [12.587156528707796]
  We introduce DAISI, a scalable filtering algorithm built on flow-based generative models.<n>We show that DAISI achieves accurate filtering results in regimes with sparse, noisy, and nonlinear observations.
  arXiv  Detail & Related papers  (2025-11-29T00:02:45Z)
• PnP-DA: Towards Principled Plug-and-Play Integration of Variational Data Assimilation and Generative Models [0.1052166918701117]
  Earth system modeling presents a fundamental challenge in scientific computing.<n>Even the most powerful AI- or physics-based forecast system suffer from gradual error accumulation.<n>We propose a Plug-and-Play algorithm that alternates a lightweight, gradient-based analysis update with a single forward pass through a pretrained prior conditioned on the background forecast.
  arXiv  Detail & Related papers  (2025-08-01T05:19:19Z)
• Decentralized Nonconvex Composite Federated Learning with Gradient Tracking and Momentum [78.27945336558987]
  Decentralized server (DFL) eliminates reliance on client-client architecture.<n>Non-smooth regularization is often incorporated into machine learning tasks.<n>We propose a novel novel DNCFL algorithm to solve these problems.
  arXiv  Detail & Related papers  (2025-04-17T08:32:25Z)
• Adaptive Federated Learning Over the Air [108.62635460744109]
  We propose a federated version of adaptive gradient methods, particularly AdaGrad and Adam, within the framework of over-the-air model training. Our analysis shows that the AdaGrad-based training algorithm converges to a stationary point at the rate of $mathcalO( ln(T) / T 1 - frac1alpha ).
  arXiv  Detail & Related papers  (2024-03-11T09:10:37Z)
• Neural Operator Variational Inference based on Regularized Stein Discrepancy for Deep Gaussian Processes [22.256068524699472]
  We introduce Neural Operator Variational Inference (NOVI) for Deep Gaussian Processes.<n>NOVI uses a neural generator to obtain a sampler and minimizes the Regularized Stein Discrepancy in L2 space between the generated distribution and true posterior.<n>We demonstrate that the bias introduced by our method can be controlled by multiplying the divergence with a constant, which leads to robust error control and ensures the stability and precision of the algorithm.
  arXiv  Detail & Related papers  (2023-09-22T06:56:35Z)
• Score-based Diffusion Models in Function Space [137.70916238028306]
  Diffusion models have recently emerged as a powerful framework for generative modeling.<n>This work introduces a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space.<n>We show that the corresponding discretized algorithm generates accurate samples at a fixed cost independent of the data resolution.
  arXiv  Detail & Related papers  (2023-02-14T23:50:53Z)
• Physics-guided Data Augmentation for Learning the Solution Operator of Linear Differential Equations [2.1850269949775663]
  We propose a physics-guided data augmentation (PGDA) method to improve the accuracy and generalization of neural operator models. We demonstrate the advantage of PGDA on a variety of linear differential equations, showing that PGDA can improve the sample complexity and is robust to distributional shift.
  arXiv  Detail & Related papers  (2022-12-08T06:29:15Z)
• Evaluating the Adversarial Robustness for Fourier Neural Operators [78.36413169647408]
  Fourier Neural Operator (FNO) was the first to simulate turbulent flow with zero-shot super-resolution. We generate adversarial examples for FNO based on norm-bounded data input perturbations. Our results show that the model's robustness degrades rapidly with increasing perturbation levels.
  arXiv  Detail & Related papers  (2022-04-08T19:19:42Z)
• Scalable Variational Gaussian Processes via Harmonic Kernel Decomposition [54.07797071198249]
  We introduce a new scalable variational Gaussian process approximation which provides a high fidelity approximation while retaining general applicability. We demonstrate that, on a range of regression and classification problems, our approach can exploit input space symmetries such as translations and reflections. Notably, our approach achieves state-of-the-art results on CIFAR-10 among pure GP models.
  arXiv  Detail & Related papers  (2021-06-10T18:17:57Z)
• Efficient Semi-Implicit Variational Inference [65.07058307271329]
  We propose an efficient and scalable semi-implicit extrapolational (SIVI) Our method maps SIVI's evidence to a rigorous inference of lower gradient values.
  arXiv  Detail & Related papers  (2021-01-15T11:39:09Z)
• Consistency analysis of bilevel data-driven learning in inverse problems [1.0705399532413618]
  We consider the adaptive learning of the regularization parameter from data by means of optimization. We demonstrate how to implement our framework on linear inverse problems. Online numerical schemes are derived using the gradient descent method.
  arXiv  Detail & Related papers  (2020-07-06T12:23:29Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.