# Accelerating Conjugate Gradient Solvers for Homogenization Problems with Unitary Neural Operators

arXiv: https://arxiv.org/abs/2508.02681 · HTML: https://mlsanity.com/paper/2508.02681/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 69 references that are in the map.
- First posted on arXiv 18 Jul 2025.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [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)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers](https://mlsanity.com/paper/2208.13273/) (2022)
- [Incremental Spatial and Spectral Learning of Neural Operators for Solving Large-Scale PDEs](https://mlsanity.com/paper/2211.15188/) (2022)
- [A numerical method for computing the overall response of nonlinear composites with complex microstructure](https://mlsanity.com/paper/2012.08962/) (2020)
- [Neural Green's Operators for Parametric Partial Differential Equations](https://mlsanity.com/paper/2406.01857/) (2024)
- [Neural Preconditioning Operator for Efficient PDE Solves](https://mlsanity.com/paper/2502.01337/) (2025)
- [Machine Learning with Physics Knowledge for Prediction: A Survey](https://mlsanity.com/paper/2408.09840/) (2024)
- [Data-Driven Microstructure Property Relations](https://mlsanity.com/paper/1903.10841/) (2019)

## Built on by (most cited first, 1 of 1)

- [NOWS: Neural Operator Warm Starts for Accelerating Iterative Solvers](https://mlsanity.com/paper/2511.02481/) (2025)

## Abstract (arXiv)

Rapid and reliable solvers for parametric partial differential equations (PDEs) are needed in many scientific and engineering disciplines. For example, there is a growing demand for composites and architected materials with heterogeneous microstructures. Designing such materials and predicting their behavior in practical applications requires solving homogenization problems for a wide range of material parameters and microstructures. While classical numerical solvers offer reliable and accurate solutions supported by a solid theoretical foundation, their high computational costs and slow convergence remain limiting factors. As a result, scientific machine learning is emerging as a promising alternative. However, such approaches often lack guaranteed accuracy and physical consistency. This raises the question of whether it is possible to develop hybrid approaches that combine the advantages of both data-driven methods and classical solvers. To address this, we introduce UNO-CG, a hybrid solver that accelerates conjugate gradient (CG) solvers using specially designed machine-learned preconditioners, while ensuring convergence by construction. As a preconditioner, we propose Unitary Neural Operators as a modification of Fourier Neural Operators. Our method can be interpreted as a data-driven discovery of Green's functions, which are then used to accelerate iterative solvers. We evaluate UNO-CG on various homogenization problems involving heterogeneous microstructures and millions of degrees of freedom. Our results demonstrate that UNO-CG enables a substantial reduction in the number of iterations and is competitive with handcrafted preconditioners for homogenization problems that involve expert knowledge. Moreover, UNO-CG maintains strong performance across a variety of boundary conditions, where many specialized solvers are not applicable, highlighting its versatility and robustness.