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

1. Problem Statement

Implicit Neural Representations (INRs), such as SIREN and NeRF, have revolutionized continuous signal representation in Euclidean domains. However, applying these methods to spherical domains ($S^2$) presents unique challenges:

• Geometric Artifacts: Standard Euclidean INRs (e.g., SIREN) ignore spherical geometry, leading to singularities and sampling artifacts, particularly near the poles.
• Computational Limitations of Existing Solutions: The state-of-the-art approach, SPH-SIREN, incorporates Spherical Harmonics (SH) into the positional encoding (PE). While effective, SPH-SIREN requires the explicit evaluation or recursive computation of SH functions. This becomes computationally expensive and numerically unstable at high harmonic orders ($L_0 > 25$), limiting scalability.

The paper aims to develop a scalable, stable, and accurate INR for spherical data that avoids explicit SH evaluations while maintaining the geometric consistency required for tasks like solving Partial Differential Equations (PDEs).

2. Methodology: Herglotz-NET (HNET)

The authors propose Herglotz-NET (HNET), a novel INR architecture that replaces standard Euclidean periodic positional encodings with a harmonic positional encoding based on Herglotz mappings.

Core Concept: Herglotz Mappings

The method leverages the Herglotz formalism, which states that for a complex vector $a \in \mathbb{C}^3$ satisfying the condition $a^\top a = 0$ (e.g., $a = (1, i, 0)$), the function $g(x) = \exp(a^\top x)$ is a harmonic function ($\Delta g = 0$) in $\mathbb{R}^3$.

• When restricted to the 2-sphere ($S^2$), the function $g(x) = \exp(\omega_0 a^\top x)$ acts as a generating function for all spherical harmonics.
• The real part of this function is centered on the real part of $a$, with oscillations oriented along the imaginary part of $a$, with frequency controlled by $\omega_0$.

Architecture

1. Positional Encoding (PE): Instead of using sine waves or explicit SHs, HNET uses a set of $n$ neurons defined by:
   $$\psi_{A,w,b}(x) = \left\{ e^{\omega_0(w_i a_i^\top x + b_i)} \right\}_{i=1}^n$$
   where $a_i$ are trainable Herglotz vectors, $w_i$ are weights, and $b_i$ are biases.
2. MLP Backbone: The encoded features are fed into a Multi-Layer Perceptron (MLP) with periodic activation functions (specifically sine, $\sin(\cdot)$), similar to SIREN.
3. Initialization: The Herglotz vectors are initialized randomly but uniformly to ensure no specific direction on the sphere is favored in expectation.

3. Key Contributions

A. Harmonic Positional Encoding without Explicit SH

HNET eliminates the need for closed-form evaluation or recursive computation of Spherical Harmonics. By using the exponential of a Herglotz mapping, the network inherently generates a basis that spans the spherical harmonic space, ensuring the representation is well-posed on the sphere and free from pole artifacts.

B. Unified Spectral Expressivity Analysis

The authors provide a theoretical analysis generalizing the spectral properties of Euclidean INRs to spherical domains.

• Lemma 1: Proves that the Taylor expansion of the Herglotz atom $(a^\top x)^\ell$ activates unique spectral components at order $\ell$ in the Spherical Harmonic basis.
• Theorem 1: Demonstrates that for any spherical INR satisfying mild conditions (polynomial approximation of activations), the spectral bandwidth scales linearly with network depth. Specifically, a network with $Q$ layers and polynomial degree $K$ can represent frequencies up to order $K^{Q-1}L_0$.
• This analysis confirms that HNET, like SPH-SIREN, possesses the capacity to represent an unbounded spectrum as depth increases.

C. Scalability and Stability

Unlike SPH-SIREN, which struggles with numerical instability at high orders, HNET relies on simple exponential and linear operations, making it computationally efficient and numerically stable even for high-frequency data.

4. Experimental Results

The authors evaluated HNET against SIREN (standard Euclidean) and SPH-SIREN (SH-based) using Earth data (EARTH2014) on Gauss-Legendre grids.

Experiment 1: Super-Resolution

• Setup: Training on a coarse grid ($L=150$) and testing on a fine grid ($L=300$).
• Results: All three models achieved nearly identical performance in terms of Signal-to-Noise Ratio (SNR $\approx 22.26$ dB).
• Observation: HNET successfully generalizes to finer resolutions without the computational overhead of explicit SH calculations.

Experiment 2: Spherical Laplacian Reconstruction

• Setup: Estimating the continuous Laplacian map of the Earth data.
• Results:
  • SIREN: Produced significant artifacts near the poles, resulting in a very low SNR ($-20.95$ dB). This indicates a failure to maintain differentiability in the spherical geometry.
  • SPH-SIREN & HNET: Both models reconstructed the Laplacian faithfully with no visible pole artifacts. HNET achieved an SNR of 7.89 dB, comparable to SPH-SIREN (6.83 dB).
• Significance: This highlights that while standard INRs can fit data, they fail in tasks requiring geometric consistency (like PDE solving), whereas HNET maintains the necessary well-posed differentiability.

5. Significance and Conclusion

Herglotz-NET represents a significant advancement in spherical deep learning by bridging the gap between the flexibility of INRs and the geometric rigor of spherical harmonics.

• Practical Impact: It offers a scalable alternative to SPH-SIREN, removing the bottleneck of explicit SH computation while matching its accuracy and stability.
• Theoretical Impact: The paper establishes a unified framework for understanding the spectral expressivity of spherical INRs, proving that depth controls frequency resolution.
• Application: HNET is particularly well-suited for applications requiring high-fidelity spherical data modeling and differentiability, such as climate modeling, global weather prediction, and solving PDEs on spherical domains, where avoiding pole singularities is critical.