Convolutional neural network-driven preconditioners for conjugate gradients

Johannes Sappl, Viktor Daropoulos, Wolfgang Rauch, Matthias Harders

Abstract We present a data-driven approach for preconditioning large sparse symmetric positive definite linear systems using convolutional neural networks tailored for sparse tensor inputs. Our work targets system matrices arising from the discretization of partial differential equations (PDE), such as those in computational fluid dynamics, which are characterized by sparsity and structured patterns. The proposed fully convolutional architecture receives the system matrix and outputs a Cholesky-like preconditioner. It is trained to approximate the inverse directly from data, without relying on traditional heuristics. We evaluate our method against no preconditioning, Jacobi, incomplete Cholesky (IC), incomplete LU, a recent graph neural network-based approach, and algebraic multigrid (AMG) as a highly specialized gold-standard reference. Experimental results demonstrate that our model achieves statistically significantly lower total solve times relative to all non-AMG baselines, including the state-of-the-art neural incomplete factorization, driven by inexpensive per-iteration application rather than by a reduced condition number. Algebraic multigrid remains faster in both case studies, consistent with its specialization for elliptic PDE. Iteration counts are reduced relative to the unpreconditioned baseline and Jacobi. Methods such as IC achieve substantially fewer iterations but at a higher per-iteration cost. Our learned preconditioner generalizes to higher-resolution problems without retraining and consistently improves system conditioning. We consider these results encouraging: convolutional architectures are a viable data-driven path toward scalable preconditioning without requiring problem-specific structural knowledge.