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Accurate Differential Operators for Hybrid Neural Fields

CVPR 2025 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Neural operators for general geometries · 2 citations (OpenAlex)

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Proposes a post hoc local polynomial-fitting differential operator and a self-supervised fine-tuning loss to make autodiff derivatives of hybrid neural fields (Instant NGP, Dense Grid, Tri-plane) accurate, evaluated on SDF normals/curvature and downstream rendering, collision, and PDE simulation.

Problem

Hybrid neural fields with explicit feature grids fit signals well but their autodiff derivatives (normals, curvature) are noisy because high-frequency grid components are amplified by differentiation, which breaks downstream rendering and simulation.

Approach

Replace direct autodiff derivatives with derivatives of a local least-squares polynomial (linear for gradients, quadratic for Hessian/Laplacian) fit to field samples drawn from a Gaussian neighborhood, and distill these smoothed derivatives back into the network via an auxiliary gradient-matching loss during fine-tuning.

Key results

• Post hoc polynomial-fitting operator reduces Instant NGP surface normal angular error from 12.40 (autodiff) to 2.80 degrees, a roughly 4x reduction versus finite differences (4.20).
• Fine-tuning with the polynomial-fit gradient loss lowers Instant NGP autodiff normal angular error from 12.40 to 3.19 degrees while Chamfer distance stays near 9.28e-4 (baseline 9.24e-4).
• In collision simulation, normals from the method give 0.85 degree error versus 11.51 degrees for autodiff normals averaged over 10^6 trials.

Measured effects reported in the paper

Surprising findings

• Eikonal regularization degraded or failed to improve autodiff normal accuracy in some settings, e.g., FD-Eikonal barely changed Instant NGP normals (12.46 degrees vs 12.40 baseline).

Limitations

• Requires choosing the smoothing scale sigma per downstream application.
• Post hoc operator needs k neighborhood forward passes per query, adding cost.
• Only removes high-frequency noise; cannot fix low-frequency errors common in non-hybrid architectures like SIREN.

Experimental setup

Datasets: FamousShape

Benchmarks: FamousShape

Metrics: L2 normal error mean angular error AA@1 AA@2 rectified relative error (curvature) Chamfer distance F-score collision normal angle error MSE for advection PDE

Baselines: Automatic differentiation Finite differences Gaussian averaged derivatives (GAD) Stochastic finite differences (SFD) Eikonal regularization FD-Eikonal Grid solver

Builds on: Instant NGP Sobolev training eikonal regularization stochastic finite differences Implicit Neural Spatial Representations

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