Investigating Hybrid Spherical Harmonic–Neural Network Models for Efficient Functional Approximation of the Global Gravity Field Representation

Global gravity fields are traditionally represented using linear combinations of spherical harmonic basis functions, whose coefficients encode the geophysical signal. While this formulation is theoretically well-founded and guarantees universal approximation on the sphere, achieving high accuracy for high-frequency features requires spherical harmonics of very high degree, leading to a rapidly increasing number of parameters and substantial computational cost. As an alternative, recent works have shown that neural networks and numerical methods can represent gravity fields by learning coordinate-to-field mappings directly, often achieving competitive accuracy with fewer parameters.

In this work, we investigate storing the gravity field implicitly as neural network weights, using spherical harmonic expansions as input to sinusoidal representation networks (SIRENs), enabling a nonlinear mapping of gravity field components. Unlike classical linear spherical harmonic models, this hybrid approach does not rely on the full harmonic basis for field reconstruction; instead, it achieves accurate representations using fewer spherical harmonics, with the network’s nonlinearity compensating for the reduced expansion. Thus, the aim of this research is to compare the performance and efficiency of this hybrid approach versus the classical spherical harmonics expansion with respect to the Earth’s gravity modeling problem. 

To create a reference ground truth to approximate, we generate high-resolution acceleration data from the EGM2008 gravity model, sampling 5,000,000 points on an equal area grid for training, and 250,000 points on a Fibonacci grid for testing; both at EGM’s reference sphere. We remove the contribution associated with planetary oblateness to isolate higher-degree features. Using this dataset, we systematically evaluate a range of model complexities.

Results show that, for lower total parameter counts, the hybrid approach achieves a lower approximation error than the standard spherical harmonic expansion. Beyond a certain model complexity, a crossover behavior is observed, after which the standard spherical harmonic expansion surpasses the hybrid representation. This is consistent with increasing representational redundancy in the nonlinear network at high model complexity, whereas the classical spherical harmonic expansion allocates parameters efficiently through its orthogonal basis functions. 

The location of this break-even point is strongly influenced by the number of input spherical harmonics. In particular, a hybrid configuration using spherical harmonics up to degree L = 15 in the first layer combined with a six-layer SIREN consistently achieves lower approximation error in the test dataset than purely linear spherical harmonic models for parameter counts up to approximately 200,000, corresponding to a linear expansion up to degree 446. For inference on 250,000 samples, the equivalent linear spherical harmonic model takes 205.872 s, while the hybrid approach requires only 0.028 s, highlighting the computational efficiency of the method. 

Compared to purely coordinate-based neural networks, the hybrid model achieves better accuracy for similar parameter budgets. These results suggest that hybrid spherical harmonic–neural models offer an attractive trade-off between accuracy, parameter efficiency, and computational cost for global gravity field modeling. This study considered only 2-dimensional fields without an explicit radial component. Extending the hybrid representation to upward continuation and evaluation of functionals in 3D space is a direction for future work.