Deep learning solvers for partial differential equations typically have limited accuracy. We propose to overcome this problem by using them as preconditioners. More specifically, we apply discretization-invariant neural operators to learn preconditioners for the flexible conjugate gradient method (FCG). Architecture paired with novel loss function and training scheme allows for learning efficient preconditioners that can be used across different resolutions. On the theoretical side, FCG theory allows us to safely use nonlinear preconditioners that can be applied in $O(N)$ operations without constraining the form of the preconditioners matrix. To justify learning scheme components (the loss function and the way training data is collected) we perform several ablation studies. Numerical results indicate that our approach favorably compares with classical preconditioners and allows to reuse of preconditioners learned for lower resolution to the higher resolution data.

翻译：深度学习求解偏微分方程通常精度有限。我们提出通过将其作为预处理器来克服这一问题。具体而言，我们采用离散化不变的神经算子来学习灵活共轭梯度法（FCG）的预处理器。结合新型损失函数与训练策略的架构，能够学习适用于不同分辨率的高效预处理器。理论方面，FCG理论允许我们安全使用非线性预处理器，此类预处理器可在$O(N)$次运算内完成，且对预处理矩阵形式无约束。为验证学习策略的各个组成部分（损失函数及训练数据采集方式），我们进行了多项消融实验。数值结果表明，我们的方法相较于经典预处理器具有显著优势，并允许将低分辨率下学习的预处理器复用于高分辨率数据。