Probabilistic inference over spatially embedded variables requires beliefs that respect $SE(3)$ symmetry, yet existing equivariant networks produce only scalars and vectors -- not the rank-2 precision tensors needed for anisotropic uncertainty, and single-component messages collapse multi-modal energy landscapes to physically meaningless averages. We introduce Equivariant Neural Belief Propagation (ENBP), a factor-graph framework whose messages are equivariant Gaussian mixture models with sufficient statistics that transform exactly under $SE(3)$. Rank-2 precision matrices are synthesised via equivariant outer products, ingested through differentiable spectral decomposition, and kept tractable by a greedy KL-based mixture reduction that provably commutes with $SE(3)$. On GEOM-QM9 and GEOM-Drugs, ENBP achieves 98.9% conformational coverage at 0.090 $\mathring{A}$ error with sub-second latency -- over $100\times$ faster than diffusion baselines at higher accuracy. On multi-body robotic inference, vanilla loopy BP diverges at 15+ agents while ENBP converges with near-zero collision rates and machine-precision equivariance error (${\sim}10^{-7}$ vs.\ $10^{-1}$ for augmented baselines).

翻译：对空间嵌入变量执行概率推断时，要求信念遵循$SE(3)$对称性，但现有等变网络仅能生成标量和向量——无法提供各向异性不确定性所需二阶精度张量，且单分量消息会将多模态能量景观坍缩为物理无意义的平均值。我们提出等变神经信念传播（ENBP），这是一种因子图框架，其消息为等变高斯混合模型，其充分统计量在$SE(3)下精确变换。二阶精度矩阵通过等变外积合成，经由可微谱分解处理，并通过基于贪婪KL散度的混合约简保持可计算性，该约简被证明与$SE(3)$可交换。在GEOM-QM9和GEOM-Drugs上，ENBP以亚秒级延迟在0.090 $\mathring{A}$误差下实现98.9%构象覆盖率——其精度高于扩散基线方法的同时快超100倍。在多体机器人推理中，标准循环置信传播在15个以上智能体时发散，而ENBP以近零碰撞率和机器精度级等变误差（${\sim}10^{-7}$对比增强基线的$10^{-1}$）收敛。