Tunable Non-linear Manifold ROMs for PDEs via Matrix-Free Petrov–Galerkin Projection

The cost of a full-order PDE solve grows quickly as the mesh is refined, and neural operators such as Fourier Neural Operators and DeepONets generally provide only one accuracy-efficiency trade-off per trained model, and adjusting it requires retraining the model. We present a nonlinear-manifold reduced-order model (NM-ROM) whose query time grows far more slowly with the mesh than the full-order solver's, so its speedup grows as the mesh is refined, and whose balance between accuracy and speed is chosen when the model is used, not when it is trained. Its spatial bank, a set of learned spatial functions, is ordered once by importance after training, so a query chooses how many bank functions to use, and whether to solve with a nonlinear head or in the linear span of the bank, without changing a learned weight. For Burgers, Poisson and heat, the trained network can be used unchanged on meshes much finer than those it was trained on, while our POD-LSPG baseline has to be rebuilt for each mesh. The model is a coordinate-network bank with exact Dirichlet enforcement, solved by least-squares Petrov–Galerkin projection of the discrete residual onto fixed tests. Every linear term is precomputed, the solver runs matrix-free in JAX, and the nonlinear term uses empirical quadrature. On viscous Burgers and three-dimensional Navier–Stokes it is up to 27.5× faster than the full-order solver, and on linear Poisson and heat problems 108× and 216× faster at 0.83% and 0.13% worst-case error. On Burgers, refining the mesh from $256^2$ to $4096^2$ raises its query time 2.6× and the solver's 74×. At the same latent dimension it is also more accurate than POD-LSPG and a quadratic manifold.