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.

翻译：参数化偏微分方程（PDE）的快速可靠求解器在众多科学与工程领域具有迫切需求。例如，对具有异质微观结构的复合材料和结构材料的需求日益增长。设计此类材料并预测其在实际应用中的行为，需要针对广泛的材料参数和微观结构求解均匀化问题。尽管经典数值求解器在坚实的理论基础上提供了可靠且精确的解决方案，但其高昂的计算成本和缓慢的收敛速度仍然是限制因素。因此，科学机器学习正成为一种有前景的替代方案。然而，这类方法通常缺乏有保证的精度和物理一致性。这引发了一个问题：是否可能开发出结合数据驱动方法与经典求解器优势的混合方法？为此，我们提出了UNO-CG，一种混合求解器，它利用专门设计的机器学习预处理器来加速共轭梯度（CG）求解器，同时通过构造确保收敛性。作为预处理器，我们提出了酉神经算子，它是对傅里叶神经算子的改进。我们的方法可被理解为格林函数的数据驱动发现，进而用于加速迭代求解器。我们在涉及异质微观结构和数百万自由度的多种均匀化问题上评估了UNO-CG。结果表明，UNO-CG能够显著减少迭代次数，并且在涉及专家知识的均匀化问题上可与手工设计的预处理器相媲美。此外，UNO-CG在多种边界条件下均保持强劲性能，而许多专用求解器在此类条件下并不适用，这凸显了其多功能性和鲁棒性。