Gravitational-wave solvers get a neural preconditioner: 2 to 15 fewer GMRES iterations than basic ILU in the hardest high-frequency regime
Synopsis
High-accuracy gravitational-wave source modeling has long avoided machine learning because neural networks carry a noise floor. Researchers trained a graph neural network as a preconditioner for an iterative solver, leaving the traditional GMRES to enforce final accuracy while the network only accelerates convergence. On the high-frequency elliptic equations for asymmetric-mass black hole binaries, the hardest parameter regions need 2 to 15 fewer GMRES iterations than basic ILU and 2 to 4 times less wall time than the next best method.
Interpretation
Machine learning is placed inside the iterative solver as an accelerator rather than predicting the solution directly, so final accuracy remains guaranteed by the traditional Krylov method. High-accuracy gravitational-wave source modeling previously relied on traditional numerical methods, and direct machine-learning applications in gravity typically saturate near a noise floor that cannot meet the sensitivity of detectors such as LISA. On the ten coupled Lorenz-gauge elliptic equations for asymmetric-mass black hole binaries, the neural preconditioner converges to the prescribed relative residual and, in the hardest high-frequency cases, uses 2 to 15 fewer GMRES iterations than basic ILU.
The graph neural network treats the sparse matrix as a weighted adjacency matrix and outputs a correction to an incomplete LU factorization, producing a more effective preconditioner. The training loss directly measures the residual reduction of each GMRES restart cycle, with gradients backpropagated by differentiating through the Arnoldi process, instead of using stochastic estimates of the Frobenius norm. Training uses a fixed resolution with about 190,000 unknown field values, 17 frequencies per azimuthal number, and validation on held-out cases.
The same preconditioning idea transfers to elastic-wave problems and still outperforms traditional iterative methods. Shear-wave components in elastodynamics defeat even standard multigrid smoothers, and a complex-shifted Laplacian preconditioner successful for the acoustic wave equation had not been extended to the elastic case. On the homogeneous half-space Lamb problem and a non-homogeneous half-space, a single network requires about half as many GMRES iterations as ILU(1).
Perspective
The demonstrated scope is limited to the first-order metric perturbation equations for asymmetric-mass black hole binaries discretized with second-order finite differences in two dimensions, plus a two-dimensional axisymmetric elastic-wave example; training uses a fixed resolution with about 190,000 unknown field values, and inference takes about 2 seconds on a laptop CPU.
The neural preconditioner is a local construction that exchanges information only between nearby grid points, so it cannot supply the global coarse-scale corrections of multigrid and the iteration count still grows with resolution; at low frequencies inference becomes a non-negligible fraction of total wall time and the advantage over ILU narrows.