# Universal Fourier Neural Operators for periodic homogenization problems in linear elasticity

arXiv: https://arxiv.org/abs/2507.12233 · HTML: https://mlsanity.com/paper/2507.12233/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 86 references that are in the map.
- First posted on arXiv 16 Jul 2025.

## Builds on (most cited first, 20 of 20)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [Error bounds for approximations with deep ReLU neural networks in Ws,p norms](https://mlsanity.com/paper/1902.07896/) (2019)
- [An overview on deep learning-based approximation methods for partial differential equations](https://mlsanity.com/paper/2012.12348/) (2020)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [Multigrid-augmented deep learning preconditioners for the Helmholtz equation](https://mlsanity.com/paper/2203.11025/) (2022)
- [Unifying the design space and optimizing linear and nonlinear truss metamaterials by generative modeling](https://mlsanity.com/paper/2306.14773/) (2023)
- [Three-dimensional convolutional neural network (3D-CNN) for heterogeneous material homogenization](https://mlsanity.com/paper/2002.07600/) (2020)
- [A numerical method for computing the overall response of nonlinear composites with complex microstructure](https://mlsanity.com/paper/2012.08962/) (2020)
- [A Spectral-based Physics-informed Finite Operator Learning for Prediction of Mechanical Behavior of Microstructures](https://mlsanity.com/paper/2410.19027/) (2024)
- [A Pretraining-Finetuning Computational Framework for Material Homogenization](https://mlsanity.com/paper/2404.07943/) (2024)
- [Learning Homogenization for Elliptic Operators](https://mlsanity.com/paper/2306.12006/) (2023)
- [UGrid: An Efficient-And-Rigorous Neural Multigrid Solver for Linear PDEs](https://mlsanity.com/paper/2408.04846/) (2024)
- [PH-Net: Parallelepiped Microstructure Homogenization via 3D Convolutional Neural Networks](https://mlsanity.com/paper/2201.09672/) (2022)
- [An FFT-based method for computing the effective crack energy of a heterogeneous material on a combinatorially consistent grid](https://mlsanity.com/paper/2103.05968/) (2021)

## Built on by (most cited first, 3 of 3)

- [Robust inverse material design with physical guarantees using the Voigt-Reuss Net](https://mlsanity.com/paper/2511.11388/) (2025)
- [Tackling multiphysics problems via finite element-guided physics-informed operator learning](https://mlsanity.com/paper/2603.01420/) (2026)
- [Nonlinear elliptic homogenization with the parametric Deep Ritz method](https://mlsanity.com/paper/2609.05778/) (2026)

## Abstract (arXiv)

Solving cell problems in homogenization is hard, and available deep-learning frameworks fail to match the speed and generality of traditional computational frameworks. More to the point, it is generally unclear what to expect of machine-learning approaches, let alone single out which approaches are promising. In the work at hand, we advocate Fourier Neural Operators (FNOs) for micromechanics, empowering them by insights from computational micromechanics methods based on the fast Fourier transform (FFT). We construct an FNO surrogate mimicking the basic scheme foundational for FFT-based methods and show that the resulting operator predicts solutions to cell problems with arbitrary stiffness distribution only subject to a material-contrast constraint up to a desired accuracy. In particular, there are no restrictions on the material symmetry like isotropy, on the number of phases and on the geometry of the interfaces between materials. Also, the provided fidelity is sharp and uniform, providing explicit guarantees leveraging our physical empowerment of FNOs. To show the desired universal approximation property, we construct an FNO explicitly that requires no training to begin with. Still, the obtained neural operator complies with the same memory requirements as the basic scheme and comes with runtimes proportional to classical FFT solvers. In particular, large-scale problems with more than 100 million voxels are readily handled. The goal of this work is to underline the potential of FNOs for solving micromechanical problems, linking FFT-based methods to FNOs. This connection is expected to provide a fruitful exchange between both worlds.