Paul Häusner

Paul Häusner, Aleix Nieto Juscafresa, Jens Sjölund

Incomplete LU factorizations of sparse matrices are widely used as preconditioners in Krylov subspace methods to speed up solving linear systems. Unfortunately, computing the preconditioner itself can be time-consuming and sensitive to hyper-parameters. Instead, we replace the hand-engineered algorithm with a graph neural network that is trained to approximate the matrix factorization directly. To apply the output of the neural network as a preconditioner, we propose an output activation function that guarantees that the predicted factorization is invertible. Further, applying a graph neural network architecture allows us to ensure that the output itself is sparse which is desirable from a computational standpoint. We theoretically analyze and empirically evaluate different loss functions to train the learned preconditioners and show their effectiveness in decreasing the number of GMRES iterations and improving the spectral properties on synthetic data. The code is available at https://github.com/paulhausner/neural-incomplete-factorization.

Paul Häusner, Ozan Öktem, Jens Sjölund

The convergence of the conjugate gradient method for solving large-scale and sparse linear equation systems depends on the spectral properties of the system matrix, which can be improved by preconditioning. In this paper, we develop a computationally efficient data-driven approach to accelerate the generation of effective preconditioners. We, therefore, replace the typically hand-engineered preconditioners by the output of graph neural networks. Our method generates an incomplete factorization of the matrix and is, therefore, referred to as neural incomplete factorization (NeuralIF). Optimizing the condition number of the linear system directly is computationally infeasible. Instead, we utilize a stochastic approximation of the Frobenius loss which only requires matrix-vector multiplications for efficient training. At the core of our method is a novel message-passing block, inspired by sparse matrix theory, that aligns with the objective of finding a sparse factorization of the matrix. We evaluate our proposed method on both synthetic problem instances and on problems arising from the discretization of the Poisson equation on varying domains. Our experiments show that by using data-driven preconditioners within the conjugate gradient method we are able to speed up the convergence of the iterative procedure. The code is available at https://github.com/paulhausner/neural-incomplete-factorization.

Ella J. Schmidtobreick, Daniel Arnström, Paul Häusner et al.

Quadratic programming (QP) solvers are widely used in real-time control and optimization, but their computational cost often limits applicability in time-critical settings. We propose a learning-to-optimize approach using graph neural networks (GNNs) to predict active sets in the dual active-set solver DAQP. The method exploits the structural properties of QPs by representing them as bipartite graphs and learning to identify the optimal active set for efficiently warm-starting the solver. Across varying problem sizes, the GNN consistently reduces the number of solver iterations compared to cold-starting, while performance is comparable to a multilayer perceptron (MLP) baseline. Furthermore, a GNN trained on varying problem sizes generalizes effectively to unseen dimensions, demonstrating flexibility and scalability. These results highlight the potential of structure-aware learning to accelerate optimization in real-time applications such as model predictive control.