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ViscoReg: Neural Signed Distance Functions via Viscosity Solutions

Meenakshi Krishnan, Ramani Duraiswami

TL;DR

This work tackles the instability and ill-posedness of learning neural signed distance functions (SDFs) under the Eikonal constraint. By grounding regularization in viscosity-solution theory, it introduces ViscoReg, a decaying viscosity term that stabilizes training and biases solutions toward the physically meaningful viscosity solution. The authors establish generalization error bounds that connect training residuals to the global error in the learned SDF, and they analyze gradient-flow dynamics under the ViscoReg energy. Empirically, ViscoReg delivers substantial gains over state-of-the-art methods on ShapeNet, the Surface Reconstruction Benchmark, and scene-reconstruction datasets, while avoiding over-smoothing of fine details. The approach provides a principled framework for neural PDE solvers and generalizes to other Hamilton–Jacobi equations."

Abstract

Implicit Neural Representations (INRs) that learn Signed Distance Functions (SDFs) from point cloud data represent the state-of-the-art for geometrically accurate 3D scene reconstruction. However, training these Neural SDFs often requires enforcing the Eikonal equation, an ill-posed equation that also leads to unstable gradient flows. Numerical Eikonal solvers have relied on viscosity approaches for regularization and stability. Motivated by this well-established theory, we introduce ViscoReg, a novel regularizer that provably stabilizes Neural SDF training. Empirically, ViscoReg outperforms state-of-the-art approaches such as SIREN, DiGS, and StEik on ShapeNet, the Surface Reconstruction Benchmark, and 3D scene reconstruction datasets. Additionally, we establish novel generalization error estimates for Neural SDFs in terms of the training error, using the theory of viscosity solutions.

ViscoReg: Neural Signed Distance Functions via Viscosity Solutions

TL;DR

This work tackles the instability and ill-posedness of learning neural signed distance functions (SDFs) under the Eikonal constraint. By grounding regularization in viscosity-solution theory, it introduces ViscoReg, a decaying viscosity term that stabilizes training and biases solutions toward the physically meaningful viscosity solution. The authors establish generalization error bounds that connect training residuals to the global error in the learned SDF, and they analyze gradient-flow dynamics under the ViscoReg energy. Empirically, ViscoReg delivers substantial gains over state-of-the-art methods on ShapeNet, the Surface Reconstruction Benchmark, and scene-reconstruction datasets, while avoiding over-smoothing of fine details. The approach provides a principled framework for neural PDE solvers and generalizes to other Hamilton–Jacobi equations."

Abstract

Implicit Neural Representations (INRs) that learn Signed Distance Functions (SDFs) from point cloud data represent the state-of-the-art for geometrically accurate 3D scene reconstruction. However, training these Neural SDFs often requires enforcing the Eikonal equation, an ill-posed equation that also leads to unstable gradient flows. Numerical Eikonal solvers have relied on viscosity approaches for regularization and stability. Motivated by this well-established theory, we introduce ViscoReg, a novel regularizer that provably stabilizes Neural SDF training. Empirically, ViscoReg outperforms state-of-the-art approaches such as SIREN, DiGS, and StEik on ShapeNet, the Surface Reconstruction Benchmark, and 3D scene reconstruction datasets. Additionally, we establish novel generalization error estimates for Neural SDFs in terms of the training error, using the theory of viscosity solutions.

Paper Structure

This paper contains 31 sections, 6 theorems, 44 equations, 7 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Surface Reconstruction
  4. Implicit Neural Representations
  5. Neural PDE solvers
  6. Error Analysis
  7. Neural Signed Distance Functions
  8. Generalization Error
  9. Proof Sketch:
  10. ViscoReg
  11. Energy Formulations and Gradient Flow
  12. ViscoReg
  13. Results
  14. Conclusion
  15. Appendix
  16. ...and 16 more sections

Key Result

Lemma 1

Let $u_1, u_2 \in C(\bar{\Omega})$ be viscosity solutions of the Eikonal equation $\|\nabla u\|_2 =f$, subject to the respective boundary conditions ${u_1}_{| \partial \Omega}=g_1$, ${u_2}_{| \partial \Omega}=g_2,$ for $g_1, g_2 \in C(\partial \Omega)$. Then:

Figures (7)

  • Figure 1: Reconstructing the 2D fractal Mandelbrot set using different Neural SDF techniques. SIREN: Converges quickly but the boundary is poorly reconstructed with many self-intersections. DiGS: Overly smoothed boundary in early iterations, with the final reconstructed boundary being disconnected and self-intersecting. StEik: While it avoids oversmoothing, it struggles with spurious self-intersections, disconnections and not capturing fine detail. ViscoReg: Smoothly converges to the underlying complex boundary, maintaining its intricate structure throughout training.
  • Figure 2: Results from the scene reconstruction benchmark from sitzmann2019scene. The DiGS mesh (a) is missing fine details like the sofa legs, accurate vase shape on the right, and picture frame details. StEik (b) performs better but struggles with fine details such as the curtains and plate on the table. The ViscoReg mesh (c) reconstructs fine details with high fidelity.
  • Figure 3: Results from ShapeNet for examples drawn from the Chair (top left), Lamp (bottom right), Car (top right) and Table (bottom right) categories. DiGS and StEik results do not maintain sharp details, and exhibit ghost pieces and other artifacts. ViscoReg mesh avoids ghost geometry and reconstructs fine surface details with high fidelity. The results are from surfaces reconstructed with StEik + quadratic layers, and DiGS, ViscoReg with standard linear layers.
  • Figure 4: Qualitative results from SRB.
  • Figure 5: Deviation from Eikonal.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Definition A.1: Viscosity Solution
  • Theorem 2
  • Theorem 3
  • Theorem 4