Hermite-NGP: Gradient-Augmented Hash Encoding for Learning PDEs

We propose Hermite-NGP, a gradient-augmented multi-resolution hash encoding designed to enable fast and accurate computation of spatial derivatives for neural PDE solvers. Unlike existing NGP-based approaches that rely on automatic differentiation or finite differences and suffer from instability or high cost, Hermite-NGP explicitly stores function values and mixed partial derivatives at hash grid vertices, allowing fully analytic evaluation of gradients, Jacobians, and Hessians via Hermite interpolation. This design preserves the efficiency and spatial adaptivity of NGP while supporting analytic differential operators up to second order. We further introduce a multi-resolution curriculum training strategy analogous to multigrid V-cycles to enable coarse-to-fine optimization. Across a range of 2D and 3D PDE benchmarks, Hermite-NGP achieves up to ${\sim}20{\times}$ lower error than prior neural PDE methods, and reduces wall-clock convergence time by $2$ – $10\times$ compared to other solvers, with per-epoch training times as low as $3.5 \mathrm{ms}$ for models with up to $17$M parameters.

Predicting how fluids swirl, how waves propagate, or how curved surfaces bend relies on equations whose answers depend on the slopes and curvatures of unknown quantities. Neural networks promise faster, more flexible alternatives to traditional physics simulators, but they keep stumbling on the same thing: computing those slopes and curvatures accurately. We built Hermite-NGP, a neural method that stores both the values of its solution and its local slopes at every grid point. With both already on hand, the network reads off the required slopes and curvatures exactly in a single pass, instead of estimating them with brittle numerical tricks. Across 2D and 3D physics benchmarks, Hermite-NGP is up to about 20 times more accurate than prior neural solvers and converges 2 to 10 times faster, while training in minutes on a single GPU.