Courant: a State-Adaptive Perceiver-Based Neural Surrogate with Local Support and Interpretable Field Decomposition

Neural surrogates promise large speedups over classical solvers for physical dynamics but fail silently at sharp dynamical events such as shocks, fronts, and contact. We present hybrid neural world models for physical dynamics: a recipe for training and deploying multi-horizon surrogates in physical state space, where a single network with continuous horizon conditioning is trained with direct supervision against textbook reference solvers to predict any future state at horizon T in one forward pass. Although no part of the training data, loss function, or architecture supervises discontinuity location, the trained surrogate encodes it implicitly, recoverable from its forward passes alone as a per-trajectory error map that concentrates on shocks, fronts, and contacts, and stays small elsewhere. The map is competitive with or better than standard label-free baselines including deep ensembles, learned error heads, gradient-magnitude indicators, and locally-adaptive conformal prediction, while using only a single trained network and requiring no calibration set or governing-equation knowledge. The recipe supports two operating points. Mode 1 runs the surrogate alone for maximum throughput, with same-hardware CPU speedups of 26x to 72x against textbook solvers on the PDE environments. Mode 2 uses the error map to gate a reference-solver fallback, deferring uncertain trajectories and roughly halving the surrogate's residual error at the default operating point. The recipe applies without modification across reaction-diffusion, compressible Euler, and rigid-body collision dynamics.

Machine-learned surrogate modeling of advection may accelerate geoscientific models, but existing approaches have either achieved limited speedup or have sacrificed spatial resolution compared to the model they are trained to emulate. We developed a machine-learned solver that speeds up advection simulations without sacrificing spatial resolution through the use of temporal coarse-graining, where the model is trained to take larger integration steps than dictated by the Courant-Friedrich-Lewy (CFL) condition. Our solver framework includes a convolutional neural network that takes concentrations and CFL numbers as inputs and outputs mass flux. Our solvers emulate 10-day ground-level horizontal advection simulations with r$^2$ values against the baseline ranging from 0.60--0.98 with temporal coarsening factors of 4 to 32 times the baseline integration time step. Speed increases and accuracy decreases with increased coarsening, with $r^2 = 0.24$ in accuracy lost for every factor of 10 gained in speed, reaching a maximum 92$\times$ speedup while maintaining $r^2 = 0.60$. We deliberately trained our solvers only on January ground-level wind data to examine their ability to generalize across seasons and vertical heights. The 4$\times$-coarsened learned solver successfully reproduces simulations over 72 vertical levels. The 8$\times$--16$\times$ solvers (but not 32$\times$) emulate most vertical levels. The learned solvers also generalize well across seasons, except for instabilities in June and October. With additional fine-tuning, these learned solvers could be appropriate for operational use where trading accuracy for speed could be advantageous, such as in screening tools, in ensemble simulations, or with data assimilation.

Recent advances in scientific machine learning provide a means of near-real-time solution to partial differential equations (PDEs), but lack the theoretical underpinnings of conventional simulators that support contemporary verification and validation. In this work, we construct data-driven reduced-order models that serve as structure-preserving, real-time surrogates. Remarkably, the exterior calculus that imposes physical conservation structure also exposes topological structure that we use to build a Gaussian process (GP) representation of uncertainty in state-flux relationships, ultimately yielding a Dirichlet-to-Neumann map for quantities of interest with closed-form expressions for posterior uncertainty. We specifically propose structure-preserving $H(\mathrm{div})$--$L^2$ subspaces of conventional Raviart--Thomas and $dgP_0$ elements prescribed by a lightweight transformer. Reduced-order dynamics consistent with this subspace are learned by posing a conservation law in which a GP describes the fluxes between volumes. This work hinges on a novel interface between mixed FEM spaces and GP regression; when training is posed as the optimal recovery problem (ORP), the resulting GP regression can be written as an optimization problem with equality constraints that impose a conservation structure, amenable to a fast Schur-complement training strategy. The trained model can then be solved in real time with closed-form estimators for boundary fluxes driven by prescribed Dirichlet data. The paper includes RKHS posterior error bounds for linear functionals to support uncertainty quantification, as well as numerical experiments demonstrating the accuracy of the posterior distribution as a surrogate for error estimation.