  Finding suitable preconditioners to accelerate iterative solution methods,
such as the conjugate gradient method, is an active area of research. In this
paper, we develop a computationally efficient data-driven approach to replace
the typically hand-engineered algorithms with neural networks. Optimizing the
condition number of the linear system directly is computationally infeasible.
Instead, our method generates an incomplete factorization of the matrix and is,
therefore, referred to as neural incomplete factorization (NeuralIF). For
efficient training, we utilize a stochastic approximation of the Frobenius loss
which only requires matrix-vector multiplications. At the core of our method is
a novel messagepassing block, inspired by sparse matrix theory, that aligns
with the objective of finding a sparse factorization of the matrix. By
replacing conventional preconditioners used within the conjugate gradient
method by data-driven models based on graph neural networks, we accelerate the
iterative solving procedure. We evaluate our proposed method on both a
synthetic and a real-world problem arising from scientific computing and show
its ability to reduce the solving time while remaining computationally
efficient.
