SE(3)-EQUIVARIANT ACOUSTIC GRAPH OPERATORS: PHYSICAL FOUNDATIONS, STEERABLE MESSAGE PASSING, AND ZERO-SHOT SPATIAL ARRAY GENERALIZATION

Spatial audio signal processing which include encompassing 3D direction-of-arrival (DOA) estimation, sound source localization, and acoustic scene reconstruction have remained fundamentally constrained by the structural rigidity of deep learning models. Standard multi-channel architectures concatenate microphone array recordings into fixed $C\times T\times F$ channel tensors, treating physical sensors as unstructured feature maps and causing catastrophic collapse under spatial rotations, translations, or topological changes in the sensor array. We present the SE(3)-Equivariant Acoustic Graph Network (SE3-AGN), a geometric deep learning framework bridging continuous physical wave mechanics with SE(3)-equivariant neural operators. By representing arbitrary multi-microphone arrays as 3D spatial graphs, SE3-AGN formulates node feature updates using steerable Clebsch–Gordan tensor products over spherical harmonics basis functions, dynamically modulated by cross-correlation phase-lag sequences and inter-sensor Euclidean distances. We introduce an adaptive, task-learned graph sparsification mechanism, physics-informed training constraints, an equivariance-consistency self-supervised pretraining objective, and a symmetry-preserving output head, and we prove mathematically that SE3-AGN guarantees exact equivariance under 3D rigid-body transformations (SE(3)) and invariance under node index permutations ($S_M$). On synthetic reverberant Room Impulse Response (RIR) simulations, SE3-AGN sets a new state-of-the-art in 3D DOA estimation (1.12${}^{\circ }$ MAE at RT60= 0.2s and 4.61${}^{\circ }$ at RT60= 0.8s), demonstrates zero-shot generalization to unseen microphone array geometries, and exhibits 10$\times$ higher sample efficiency over non-equivariant baselines, with real-world evaluation on the LOCATA benchmark confirming robustness across dynamic multi-speaker scenarios.