Read, Write, Relax: Why Neural PDE Surrogates Need Both Global and Local Processing

Recent mesh-based simulation advances have, in no small part, relied on neural surrogates of two distinct families: global models that route information through a small set of latent tokens, and local models that perform message passing across mesh edges. Consistent with both classes is the inability to perform beyond low-dimensional problems and small-scale or oversimplified meshes, the simulation regimes where industrial problems reside. Our work shows this explicitly and presents a unified formulation. In global approaches, latent-token attention acts as a spatial low-pass filter, while local message passing lacks the global reach necessary to propagate information across large mesh spaces. Viewed through the error, the two operators are the halves of a multigrid cycle: one corrects errors at the lower end of the spectrum, the other at the higher end, and neither can do the other's job. We introduce Read-Write-Relax (RWR), which interleaves latent attention with message-passing relaxation under a unified formulation. The interleaved processor lowers error across the entire spectrum, making RWR the most accurate model in nearly every comparison across our industrial and public benchmarks. It is also markedly data-efficient in the scarce-data regimes, accurate on the engineering quantities of interest, and scales full-field predictions to challenging, large-scale problems.

McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation

Solving heterogeneous Helmholtz equations at high wavenumbers remains challenging because the discretized operator is indefinite, pollution degrades phase accuracy, and scalar coarse-grid correction can discard the local phase and propagation-direction information carried by oscillatory errors. We propose Multi-channel Multigrid (McMg), a learned phase-space multigrid preconditioner for heterogeneous Helmholtz equations. Rather than predicting the solution directly, McMg maps residuals to corrections within an iterative framework. Its central idea is to coarsen physical space while retaining unresolved local wave information in the channel dimension: each coarse node carries a learned packet of amplitude, phase, direction, and scattering coefficients rather than a single scalar unknown. The architecture combines linear multi-channel transfer operators with locally adaptive stencils, neural PDE operators, and medium-dependent smoothers whose coefficients are generated from the wave speed. For a fixed medium, the V-cycle is linear in the residual; nonlinear physical features are computed once in a setup phase and cached, so each online iteration reduces to convolutions with fixed coefficients. We further study generalization across scales. Models trained on small domains transfer directly to larger domains and higher effective wavenumbers, and a Layer-by-Layer Progressive Finetuning (LLPF) strategy extends the support of the learned Green's operator by adding and finetuning only new coarse levels. Numerical experiments on high-frequency, high-contrast, and large-scale three-dimensional problems demonstrate that McMg requires substantially fewer iterations and less wall-clock time than strong classical baselines, while consistently outperforming existing neural preconditioners.