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

2026-05-23

AI Summary

This work addresses the instability and high computational cost of traditional neural PDE solvers that rely on automatic or finite differences for spatial derivatives. The authors propose a gradient-augmented multi-resolution hash encoding method that explicitly stores function values and mixed partial derivatives at hash grid vertices, enabling analytical computation of gradients, Jacobians, and Hessians via Hermite interpolation. A curriculum training strategy inspired by multigrid V-cycles is introduced to progressively refine solutions from coarse to fine scales. This approach is the first to integrate analytical high-order differential operators into a hash encoding framework, preserving the efficiency and spatial adaptivity of Neural Geometric Processing (NGP) while enabling fast and accurate second-order derivative evaluation. Experiments demonstrate up to a 20× reduction in error and 2–10× faster convergence across multiple 2D/3D PDE benchmarks, with single-iteration training as fast as 3.5 milliseconds for models containing 17 million parameters.

Abstract

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 approximately 20 times lower error than prior neural PDE methods, and reduces wall-clock convergence time by 2 to 10 times compared to other solvers, with per-epoch training times as low as 3.5 ms for models with up to 17M parameters.

Problem

• neural PDE solvers
• spatial derivatives
• automatic differentiation
• finite differences
• numerical instability

Innovation

• Hermite interpolation
• hash encoding
• neural PDE solvers
• analytic derivatives
• multi-resolution training