用梯度增强的哈希编码，让神经PDE求解更快更准。
Hermite-NGP: Gradient-Augmented Hash Encoding for Learning PDEs
- 在哈希网格顶点显式存储函数值与混合偏导数，实现解析求导
- 2D/3D PDE测试中误差降低约20倍，收敛时间快2~10倍
- 适合需要高精度微分的科学计算场景，如流体模拟、物理建模
我们提出Hermite-NGP，一种梯度增强的多分辨率哈希编码，用于神经PDE求解器中快速准确地计算空间导数。不同于依赖自动微分或有限差分的现有NGP方法（存在不稳定或高成本问题），Hermite-NGP在哈希网格顶点显式存储函数值和混合偏导数，通过赫米特插值实现梯度、雅可比和海森矩阵的完全解析计算。该设计保持了NGP的高效性与空间自适应性，同时支持二阶以内的解析微分算子。我们进一步引入类多重网格V型循环的多分辨率课程训练策略，实现从粗到细的优化。在多个2D与3D PDE基准测试中，Hermite-NGP相比先前神经PDE方法误差降低约20倍，壁钟收敛时间减少2至10倍，单次训练耗时低至3.5毫秒（模型参数量达1700万）。
原文摘要 · Abstract (English)
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.
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