pith. sign in

arxiv: 2605.24774 · v1 · pith:FPFXDGODnew · submitted 2026-05-23 · 💻 cs.LG · physics.comp-ph

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

Jinjin He , Zhiqi Li , Sinan Wang , Bo Zhu This is my paper

Pith reviewed 2026-06-30 13:18 UTC · model grok-4.3

classification 💻 cs.LG physics.comp-ph
keywords neural PDE solvershash encodingHermite interpolationgradient augmentationanalytic derivativesmulti-resolution trainingspatial derivativesPDE benchmarks
0
0 comments X

The pith

Hermite-NGP augments hash encodings by storing mixed partial derivatives at vertices to enable analytic gradient computation for neural PDE solvers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that standard neural PDE methods suffer from unstable or costly derivative estimates when using automatic differentiation or finite differences on hash encodings. Hermite-NGP counters this by storing both function values and mixed partial derivatives directly at the hash grid vertices. Hermite interpolation then supplies exact first- and second-order operators without extra differentiation steps. A coarse-to-fine curriculum training schedule further stabilizes optimization across resolutions. The resulting models reach substantially lower errors and converge faster on 2D and 3D benchmark problems than earlier neural approaches.

Core claim

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. Combined with a multi-resolution curriculum training strategy analogous to multigrid V-cycles, the method 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.

What carries the argument

Gradient-augmented multi-resolution hash encoding that stores function values together with mixed partial derivatives at vertices and applies Hermite interpolation for analytic derivative evaluation.

If this is right

  • Analytic derivatives remove the instability previously caused by automatic differentiation or finite differences in neural PDE training.
  • The encoding supports efficient computation of first- and second-order differential operators while retaining NGP's spatial adaptivity.
  • The multi-resolution curriculum enables stable coarse-to-fine optimization analogous to multigrid V-cycles.
  • Models with up to 17 million parameters can be trained at per-epoch times as low as 3.5 ms.
  • Error reductions and speed-ups hold across multiple 2D and 3D PDE benchmarks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Memory cost per vertex rises because additional derivative components must be stored, which may constrain grid resolution on memory-limited hardware.
  • The same storage-and-interpolation pattern could be applied to tasks outside PDEs that need reliable higher-order derivatives, such as sensitivity analysis or shape optimization.
  • Performance on problems containing discontinuities would test whether Hermite interpolation introduces unwanted smoothing.

Load-bearing premise

Storing and interpolating mixed partial derivatives at hash vertices will remain numerically stable and will not introduce new artifacts during network optimization.

What would settle it

On any of the reported 2D or 3D PDE benchmarks, observing that derivative fields computed during training exhibit growing oscillations or produce final errors higher than those of a baseline NGP method using automatic differentiation.

Figures

Figures reproduced from arXiv: 2605.24774 by Bo Zhu, Jinjin He, Sinan Wang, Zhiqi Li.

Figure 1
Figure 1. Figure 1: SDF Gradients and Curvature on Armadillo. Our method (left) recovers smoother gradient and curvature fields than NeuralAngelo (Li et al., 2023b) (right), used here as an FD-based hash-encoding baseline under the same SDF+gradient objective. Gradients are visualized as green line segments around the surface, while curvature is shown via mesh coloring. et al., 2025; Zou et al., 2024), anti-aliased NeRF vari￾… view at source ↗
Figure 2
Figure 2. Figure 2: Hermite-NGP Workflow. Multi-resolution grids store Hermite coefficients (f, fx, fy, fxy) in separate hash tables with sizes T1 (values), T2 (first derivatives), and T3 (mixed derivatives), producing encoding γ, ∇γ, ∇2 γ. The MLP outputs u, ∂u/∂γ, ∂2u/∂γ2 , combined via chain rule to yield ∇u, ∇2u for PDE loss. tion, where velocity gradients enable higher-order advection schemes. Recent neural representatio… view at source ↗
Figure 3
Figure 3. Figure 3: Gradient-augmented illustration. The left cell without gradients yields constant interpolation, while the others use gradi￾ents to produce rich sub-grid features. Hermite interpolant is: H(t) = f0h (0)(t) + f1h (0)(1 − t) + f ′ 0h (1)(t) + f ′ 1h (1)(t), h (0)(t) = ( −2t 3 + 3t 2 , 0 ≤ t ≤ 1, 0, otherwise, , h (1)(t) = ( 2t 3 − 3t 2 + 1, 0 ≤ t ≤ 1, 0, otherwise, (2) where h (i) , i ∈ {0, 1}, denote the bas… view at source ↗
Figure 4
Figure 4. Figure 4: Helmholtz 3D (a=10). Cross-sectional slices of the 3D Helmholtz solution. Hermite-NGP accurately captures the oscillatory wave patterns, achieving an L 2 error of 6 × 10−3 , substantially better than the closest baseline, I-NGP-FD (7.21 × 10−2 ). PINN SPINN JAX-PI INGP-FD PIRATENET OURS GT Prediction Error [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Helmholtz 2D (a = 100) enlarged: Hermite-NGP is the only method that converges on this challenging high-frequency setting, achieving L 2 = 4.59×10−2 . All baseline methods fail to capture the rapid oscillations. pled dynamics, complex geometries, and geometric con￾straints. For problems without analytic solutions, we obtain ground truth from high-resolution conventional PDE solvers. All metrics are reporte… view at source ↗
Figure 9
Figure 9. Figure 9: Helmholtz 2D (a=10) qualitative comparison across Hermite-NGP, ∂ ∞-Grid (Kairanda et al., 2026), SIREN. Only Hermite-NGP recovers the high-frequency structure cleanly [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: Flow Mixing: Solution field showing rotational trans￾port. Hermite-NGP achieves L 2 = 2.35×10−4 , resolving sharp gradients from fluid stretching. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: SDF Learning and Curvature Estimation. Hermite￾NGP learns an SDF whose gradients (green line segments) and cur￾vature (shown as color) remain smooth thanks to analytic second derivatives, whereas NeuralAngelo’s finite-difference curvature is noisy and exhibits visible artifacts. PIG (2.67×10−4 ) and achieving a 12× improvement over SPINN; the PIG and SPINN numbers are taken from their original publication… view at source ↗
Figure 13
Figure 13. Figure 13: Image reconstruction from gradient supervision on the camera image. Hermite-NGP recovers sharper edges and finer texture than ∂ ∞-Grid and SIREN while matching their soft-region quality. 7.6. GPU Memory and Speed Comparison Hermite-NGP scales efficiently: 68K to 16.8M parameters uses 133–389 MB memory and 1.8–3.6 ms/epoch ( [PITH_FULL_IMAGE:figures/full_fig_p008_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: 3D Poisson with mesh boundary (Armadillo). Solution u with Dirichlet values u=1 on the mesh surface and u=0 on the outer domain boundary. Hermite-NGP achieves an L 2 error of 5 × 10−3 , a 3× improvement over PIG. The third row shows L2 error. noisy features amplifies the noise. Hermite-NGP instead treats the derivative field as a first-class optimization target, distributing the representation burden acro… view at source ↗
Figure 15
Figure 15. Figure 15: Approximate Ablation. Comparison of encoding variants on Helmholtz (a = 10). Trilinear, Cubic, and Bicubic NGP all fail (L2 > 0.1), while Hermite-NGP achieves 1.81e-5, confirming stored derivative coefficients are essential [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Flow Mixing. Solution snapshots at t = 0, 0.2, 0.4, 0.6, 0.8, 1.0. Hermite-NGP achieves average L2 error of 2.35e-4 across all time steps. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Helmholtz 3D (a = 3). Cross-sectional slices of the 3D Helmholtz solution. Hermite-NGP achieves L2 error of 6.09e-5, compared to 8.40e-4 for PirateNet (13.8× lower) [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Helmholtz 2D (a = 20). High-frequency solution with 20 wavelengths. Hermite-NGP achieves 9.87e-5 (1.07M params) and 7.93e-5 (4.21M params), outperforming PirateNet (1.36e-3), JAX-PI (1.2e-3), INGP-FD (2.77e-3), and SPINN (3.30e-2) [PITH_FULL_IMAGE:figures/full_fig_p026_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Helmholtz 2D (a = 100): Hermite-NGP is the only method that converges on this challenging high-frequency setting, achieving L 2 = 4.59×10−2 . All baseline methods fail to capture the rapid oscillations [PITH_FULL_IMAGE:figures/full_fig_p026_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Training loss curves across three benchmarks. The horizontal axis indicates the number of epochs, while the vertical axis reports the relative ℓ2 training error. Our method exhibits a smoothly decreasing loss trajectory, in contrast to the strongly oscillatory behavior of the baselines, and converges to training errors up to four orders of magnitude smaller across the benchmarks; see [PITH_FULL_IMAGE:fig… view at source ↗
Figure 21
Figure 21. Figure 21: Poisson 3D (Bunny). Cross-sectional slice with mesh boundary and zoom-in detail. Hermite-NGP achieves MAE 0.0044 vs. PIG’s 0.0127 (2.9× lower), with more accurate field near the boundary. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Poisson 3D (Fandisk). Cross-sectional slice with mesh boundary. Hermite-NGP achieves MAE 0.0031 vs. PIG’s 0.0100 (3.2× lower), accurately capturing sharp geometric features [PITH_FULL_IMAGE:figures/full_fig_p028_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: SDF Curvature (Armadillo). Mesh colored by mean curvature, green lines show gradient direction. Hermite-NGP achieves gradient MAE 0.0478 vs. NeuralAngelo’s 0.1009 (2.1× lower) [PITH_FULL_IMAGE:figures/full_fig_p028_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: SDF Curvature (Lucy). Mesh colored by mean curvature, green lines show gradient direction. Hermite-NGP achieves gradient MAE 0.0418 vs. NeuralAngelo’s 0.1213 (2.9× lower). 28 [PITH_FULL_IMAGE:figures/full_fig_p028_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: SDF Curvature (Fandisk). Mesh colored by mean curvature, green lines show gradient direction. Hermite-NGP achieves gradient MAE 0.0516 vs. NeuralAngelo’s 0.1064 (2.1× lower), preserving sharp edges. Neuralangelo Ours GT [PITH_FULL_IMAGE:figures/full_fig_p029_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: SDF Curvature (Bunny). Mesh colored by mean curvature, green lines show gradient direction. Hermite-NGP achieves gradient MAE 0.0416 vs. NeuralAngelo’s 0.0887 (2.1× lower). 29 [PITH_FULL_IMAGE:figures/full_fig_p029_26.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces Hermite-NGP, a gradient-augmented multi-resolution hash encoding for neural PDE solvers. It explicitly stores function values and mixed partial derivatives at hash grid vertices to support analytic evaluation of gradients, Jacobians, and Hessians via Hermite interpolation, avoiding automatic differentiation or finite differences. A multi-resolution curriculum training strategy is proposed. On 2D and 3D PDE benchmarks, the method is reported to achieve up to approximately 20 times lower error than prior neural PDE methods and 2 to 10 times faster wall-clock convergence, with per-epoch times as low as 3.5 ms for models up to 17M parameters.

Significance. If the performance claims hold under rigorous validation, the work addresses a key practical limitation in neural PDE solvers by enabling stable, low-cost analytic derivatives while retaining the spatial adaptivity of hash encodings. This could meaningfully advance scientific machine learning applications requiring accurate higher-order derivatives.

major comments (2)
  1. [§5] §5 (Experiments): The central performance claims of up to 20x error reduction and 2-10x faster convergence are load-bearing for the contribution, yet the manuscript provides no error bars, number of independent runs, or ablation studies isolating the contribution of explicit mixed-partial storage versus the curriculum strategy. This prevents assessment of whether gains are robust or sensitive to post-hoc choices.
  2. [§3.2] §3.2 (Encoding design): The assumption that explicitly storing and Hermite-interpolating mixed partial derivatives remains numerically stable under network optimization is asserted but not supported by dedicated stability analysis, condition-number bounds, or counter-example tests; this is load-bearing for the claim of artifact-free analytic operators.
minor comments (1)
  1. [Abstract] The abstract would benefit from naming the specific PDE classes (e.g., Poisson, Navier-Stokes) used in the benchmarks for immediate context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and will revise the paper accordingly to strengthen the presentation of results and supporting analysis.

read point-by-point responses

  1. Referee: [§5] §5 (Experiments): The central performance claims of up to 20x error reduction and 2-10x faster convergence are load-bearing for the contribution, yet the manuscript provides no error bars, number of independent runs, or ablation studies isolating the contribution of explicit mixed-partial storage versus the curriculum strategy. This prevents assessment of whether gains are robust or sensitive to post-hoc choices.

    Authors: We agree that the absence of error bars, multiple runs, and targeted ablations limits evaluation of robustness. In the revised manuscript we will report all benchmark results as means and standard deviations over at least five independent random seeds, and we will add ablation experiments that separately disable explicit mixed-partial storage and the multi-resolution curriculum while keeping all other factors fixed. revision: yes

  2. Referee: [§3.2] §3.2 (Encoding design): The assumption that explicitly storing and Hermite-interpolating mixed partial derivatives remains numerically stable under network optimization is asserted but not supported by dedicated stability analysis, condition-number bounds, or counter-example tests; this is load-bearing for the claim of artifact-free analytic operators.

    Authors: We acknowledge that a dedicated stability study is needed to substantiate the numerical claims. The revision will include a new subsection in §3.2 that reports (i) condition-number bounds for the local Hermite interpolation matrices, (ii) empirical monitoring of gradient and Hessian accuracy throughout training on the benchmark problems, and (iii) targeted counter-example tests on functions known to challenge high-order interpolants. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces Hermite-NGP as a design choice that explicitly stores function values and mixed partial derivatives at hash vertices to enable analytic Hermite interpolation of derivatives up to second order, combined with a multi-resolution curriculum. No equations, predictions, or first-principles derivations are presented that reduce claimed performance gains to fitted quantities or inputs by construction. The empirical benchmarks on 2D/3D PDEs are external comparisons, and the method's internal consistency does not rely on self-citations or ansatzes that collapse to the target result. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides insufficient detail to enumerate free parameters, axioms, or invented entities; the central claim rests on the unstated assumption that Hermite interpolation of stored derivatives remains stable across the tested PDEs.

pith-pipeline@v0.9.1-grok · 5718 in / 1089 out tokens · 23744 ms · 2026-06-30T13:18:34.103629+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

9 extracted references · 4 canonical work pages · 1 internal anchor

  1. [1]

    2d gaussian splatting for geometrically accurate radiance fields

    Huang, B., Yu, Z., Chen, A., Geiger, A., and Gao, S. 2d gaussian splatting for geometrically accurate radiance fields. InACM SIGGRAPH 2024 conference papers, pp. 1–11,

    2024

  2. [2]

    Dinf-grid: A neural dif- ferential equation solver with differentiable feature grids

    Kairanda, N., Naik, S., Habermann, M., Sharma, A., Theobalt, C., and Golyanik, V . Dinf-grid: A neural dif- ferential equation solver with differentiable feature grids. arXiv preprint arXiv:2601.10715,

    work page arXiv

  3. [3]

    Pig: Physics- informed gaussians as adaptive parametric mesh repre- sentations.arXiv preprint arXiv:2412.05994,

    Kang, N., Oh, J., Hong, Y ., and Park, E. Pig: Physics- informed gaussians as adaptive parametric mesh repre- sentations.arXiv preprint arXiv:2412.05994,

    work page arXiv

  4. [4]

    Kingma, D. P. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980,

    work page internal anchor Pith review Pith/arXiv arXiv

  5. [5]

    Nerfstudio: A modular framework for neural radiance field development

    Tancik, M., Weber, E., Ng, E., Li, R., Yi, B., Wang, T., Kristoffersen, A., Austin, J., Salahi, K., Ahuja, A., et al. Nerfstudio: A modular framework for neural radiance field development. InACM SIGGRAPH 2023 conference proceedings, pp. 1–12,

    2023

  6. [6]

    An Expert's Guide to Training Physics- informed Neural Networks,

    Wang, S., Sankaran, S., Wang, H., and Perdikaris, P. An ex- pert’s guide to training physics-informed neural networks. arXiv preprint arXiv:2308.08468, 2023a. Wang, S., Li, B., Chen, Y ., and Perdikaris, P. Piratenets: Physics-informed deep learning with residual adaptive networks.Journal of Machine Learning Research, 25 (402):1–51, 2024b. Wang, S., Sanka...

    work page arXiv

  7. [7]

    The exact solution: u∗(t, x, y) =−tanh y 2 cos(ωt)− x 2 sin(ωt) .(42) This problem develops sharp gradients as fluid parcels stretch and fold under the mixing dynamics. For both Poisson 3D and SDF learning experiments, we evaluate on five Stanford meshes of varying geometric complexity: •Armadillo: High-genus mesh with intricate surface details •Bunny: Cl...

    2040

  8. [8]

    forgetting

    Loss Balancing.We use an adaptive gradient-based loss balancing scheme inspired by GradNorm (Chen et al., 2018). At each iteration, we compute the gradient norms of the PDE residual loss ( gpde) and boundary condition loss (gbc). When gbc >10 −8, the BC weight is updated via exponential moving average: λbc ←0.9·λ bc + 0.1· gpde gbc , λ bc ∈[1, λ max],(48)...

    2018

  9. [9]

    Table 8.Architecture ablation on Helmholtz 2D (a= 20)

    This suggests high-frequency solutions have finer spatial structure requiring more hash capacity. Table 8.Architecture ablation on Helmholtz 2D (a= 20). Top 10 configurations by L2 error. Hash Scale Lvls Params BestL 2 FinalL 2 16 2.0 8 4.21M 7.93e-5 6.97e-5 14 1.5 8 1.07M 3.33e-4 2.65e-4 14 2.0 6 805K 3.37e-4 2.89e-4 14 2.3 8 1.07M 5.00e-4 4.40e-4 12 2.0...

    2024