Alioth: Neural Modulators for Scientific Computing

Scientific computing underpins modern science and engineering, and solving partial differential equations (PDEs) lies at its core. For decades, the field has pursued PDE solvers that are both accurate and generalizable. However, existing approaches rarely achieve both. Classical numerical algorithms guarantee accuracy but solve each problem anew, whereas neural operators generalize but cannot guarantee accuracy. More recently, neural preconditioners embed learning into numerical solvers but typically train for each problem or problem family. In this work, we introduce the neural modulator, a single pretrained network that generates solver components across problems and solvers. We present Alioth, a neural modulator that generates coarse spaces, the solver components that capture slow-to-converge error modes. Coarse spaces are transferable, as these modes follow local operator structure shared across PDEs, and error-tolerant, as imperfect ones only slow convergence. Pretrained once on synthetic diffusion problems with a label-free spectral objective, Alioth serves domain decomposition and algebraic multigrid, the two workhorse solvers for large-scale PDEs. With domain decomposition on 13 scalar SPD systems spanning reservoir simulation, heat transfer, fluid dynamics, structural mechanics, and electromagnetics, it needs 48% fewer iterations than the classical Nicolaides coarse space and 20% fewer than unlearned subspace iteration with the same number of matrix products. Because accuracy is set by the solver, five iterations on Darcy flow reach errors over 100× below neural operators and PDE foundation models, in and out of their training distribution. These results point to a new direction for AI-enabled scientific computing: learn the solver, not the solution.