EqGINO: Equivariant Geometry-Informed Fourier Neural Operators for 3D PDEs

2026-06-02

AI Summary

This work addresses the limited generalization of existing deep learning methods for solving three-dimensional partial differential equations, which are often constrained by fixed coordinate systems and struggle with arbitrary geometric transformations. The authors propose a spectral-domain equivariant modeling framework that efficiently incorporates full SE(3) equivariance into Fourier neural operators for the first time. By introducing structural priors that respect discrete symmetries, the method avoids computationally expensive spectral group convolutions while preserving both global receptive fields and geometric robustness. Requiring only a few transformed samples, the approach generalizes effectively to continuous poses and achieves high-accuracy, coordinate-independent modeling of physical laws on complex, irregular 3D geometries, significantly outperforming current state-of-the-art methods.

Abstract

Deep learning surrogates for 3D Partial Differential Equations (PDEs) often fail to generalize across geometric transformations because they depend heavily on specific coordinate systems. While equivariant networks offer a solution, they typically rely on local operations in the spatial domain, making the global receptive field, which is essential for PDE dynamics, computationally expensive. Conversely, Fourier Neural Operators (FNOs) efficiently capture global interactions, yet establishing 3D equivariance within them remains impractical due to the prohibitive cost of spectral group convolutions. To bridge this gap, we introduce EqGINO, a geometrically robust framework that enforces isotropy in the spectral domain. By design, EqGINO guarantees exact equivariance to the discrete symmetries inherent to the discretized computational domain. Beyond this discrete guarantee, our structural prior enables effective generalization to arbitrary continuous orientations even with a limited number of SE(3)-transformed training samples. Consequently, our method robustly models coordinate-invariant physical laws on complex irregular 3D geometries. Our code is available at https://github.com/sung-won-kim/EqGINO

Problem

• 3D PDEs
• equivariance
• geometric generalization
• Fourier Neural Operators
• coordinate invariance

Innovation

• Equivariant Neural Networks
• Fourier Neural Operators
• 3D PDEs
• Spectral Domain Isotropy
• Geometric Generalization