Bridging Equivariant GNNs and Spherical CNNs for Structured Physical Domains

Many modeling tasks from disparate domains can be framed the same way, computing spherical signals from geometric inputs, for example, computing the radar response of different objects or navigating through an environment. This paper introduces G2Sphere, a general method for mapping object geometries to spherical signals. G2Sphere operates entirely in Fourier space, encoding geometric structure into latent Fourier features using equivariant neural networks and outputting the Fourier coefficients of the continuous target signal, which can be evaluated at any resolution. By utilizing a hybrid GNN-spherical CNN architecture, our method achieves much higher frequency output signal than comparable equivariant GNNs and avoids hand-engineered geometry features used previously by purely spherical methods. We perform experiments on various challenging domains including radar response modeling, aerodynamic drag prediction, and policy learning for manipulation and navigation. We find that G2Sphere outperforms competitive baselines in terms of accuracy and inference time, and we demonstrate that equivariance and Fourier features lead to improved sample efficiency and generalization.The source code is available at: https://github.com/ColinKohler/geometry2sphere.