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Herglotz-NET: Implicit Neural Representation of Spherical Data with Harmonic Positional Encoding

Théo Hanon, Nicolas Mil-Homens Cavaco, John Kiely, Laurent Jacques

TL;DR

Herglotz-NET (HNET) introduces a spherical implicit neural representation that replaces Euclidean periodic encodings with a harmonic positional encoding based on complex Herglotz mappings, yielding a well-posed representation on $\,\mathbb{S}^2$ without explicit spherical harmonic evaluations. A unified expressivity analysis shows that spherical INRs with a mild condition admit a spectral expansion whose bandwidth grows with network depth, enabling rich frequency representations. Empirical results on Earth data demonstrate that HNET matches SPH-SIREN in super-resolution and Laplacian reconstruction tasks, while avoiding pole artifacts and benefiting from stable, well-posed differentiability. The work highlights a practical, scalable approach for accurate spherical data modeling with potentially broad implications for PDE solving and inverse problems on curved domains.

Abstract

Representing and processing data in spherical domains presents unique challenges, primarily due to the curvature of the domain, which complicates the application of classical Euclidean techniques. Implicit neural representations (INRs) have emerged as a promising alternative for high-fidelity data representation; however, to effectively handle spherical domains, these methods must be adapted to the inherent geometry of the sphere to maintain both accuracy and stability. In this context, we propose Herglotz-NET (HNET), a novel INR architecture that employs a harmonic positional encoding based on complex Herglotz mappings. This encoding yields a well-posed representation on the sphere with interpretable and robust spectral properties. Moreover, we present a unified expressivity analysis showing that any spherical-based INR satisfying a mild condition exhibits a predictable spectral expansion that scales with network depth. Our results establish HNET as a scalable and flexible framework for accurate modeling of spherical data.

Herglotz-NET: Implicit Neural Representation of Spherical Data with Harmonic Positional Encoding

TL;DR

Herglotz-NET (HNET) introduces a spherical implicit neural representation that replaces Euclidean periodic encodings with a harmonic positional encoding based on complex Herglotz mappings, yielding a well-posed representation on without explicit spherical harmonic evaluations. A unified expressivity analysis shows that spherical INRs with a mild condition admit a spectral expansion whose bandwidth grows with network depth, enabling rich frequency representations. Empirical results on Earth data demonstrate that HNET matches SPH-SIREN in super-resolution and Laplacian reconstruction tasks, while avoiding pole artifacts and benefiting from stable, well-posed differentiability. The work highlights a practical, scalable approach for accurate spherical data modeling with potentially broad implications for PDE solving and inverse problems on curved domains.

Abstract

Representing and processing data in spherical domains presents unique challenges, primarily due to the curvature of the domain, which complicates the application of classical Euclidean techniques. Implicit neural representations (INRs) have emerged as a promising alternative for high-fidelity data representation; however, to effectively handle spherical domains, these methods must be adapted to the inherent geometry of the sphere to maintain both accuracy and stability. In this context, we propose Herglotz-NET (HNET), a novel INR architecture that employs a harmonic positional encoding based on complex Herglotz mappings. This encoding yields a well-posed representation on the sphere with interpretable and robust spectral properties. Moreover, we present a unified expressivity analysis showing that any spherical-based INR satisfying a mild condition exhibits a predictable spectral expansion that scales with network depth. Our results establish HNET as a scalable and flexible framework for accurate modeling of spherical data.

Paper Structure

This paper contains 8 sections, 3 theorems, 13 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. An alternative spherical INR on $\mathbb S^2$
  4. Spectrum Expansion of Spherical INRs
  5. Numerical experiments
  6. Super-Resolution
  7. Laplacian Reconstruction
  8. Conclusion

Key Result

Lemma 1

Given a vector $\boldsymbol a\in\mathbb{C}^3$, with $\boldsymbol a^\top \boldsymbol a=0$ and $\ell \in \mathbb N$ and $\|\boldsymbol a\|_2 = 1$, the function $h_\ell(\boldsymbol x)=(\boldsymbol a^\top \boldsymbol x)^\ell$ is harmonic in $\mathbb R^3$ and its restriction to all $\boldsymbol x = \bold with $c_{\ell m}(\boldsymbol a)\in \mathbb C$ such that $\sum_{|m|\leqslant \ell} |c_{\ell m}|^2 \l

Figures (4)

  • Figure 1: Representation of the real part of the Herglotz atom $g(\boldsymbol x)$ for $\boldsymbol a = (1,1,1)/\sqrt{3} + \mathrm{i}\mkern1mu (1,-1,0)/\sqrt{2}$, and different values of $\omega_0$. The function $g$ is centered on $\boldsymbol a_{\Re} = (1,1,1)/\sqrt{3}$ with oscillations locally oriented along the direction $\boldsymbol a_{\Im} = (1,-1,0)/\sqrt{2}$ and frequency proportional to $\omega_0$.
  • Figure 2: The Herglotz-NET architecture using the $n$-neuron PE defined by \ref{['eq:PE-hnet']}.
  • Figure 3: (left) Earth representation on the fine grid ($L = 300$). (right) Difference between this representation and HNET upsampled on the same grid.
  • Figure 4: Comparison of the Laplacian of the Earth representation computed with spherical harmonics up to $L = 150$ (left) and the Laplacian of the models HNET, SIREN, and SPH-SIREN (from left to right). The ground truth Laplacian is computed using the spherical harmonic transform, while the model Laplacians are obtained via automatic differentiation of the continuous functions.

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2
  • proof