让神经网络推理保持空间对称性，提升分子构象与机器人定位精度。
Equivariant Neural Belief Propagation
- 用可学习的等变高斯混合消息建模不确定性，支持复杂空间关系。
- 在分子数据集上达到98.9%构象覆盖率，误差仅0.090 Å，速度快100倍以上。
- 适用于需要精确空间对称性的场景，如分子结构预测和多智能体协同定位。
对空间嵌入变量进行概率推断需保持SE(3)对称性，但现有等变网络仅输出标量和向量，无法表示各向异性不确定性所需的二阶精度张量；单成分消息会将多模态能量景观压缩为无物理意义的平均值。本文提出等变神经信念传播（ENBP），其消息为等变高斯混合模型，充分统计量严格遵循SE(3)变换。通过等变外积生成二阶精度矩阵，经可微谱分解处理，并以基于KL的贪心混合物压缩策略保持计算可追踪性，且该压缩操作与SE(3)可交换。在GEOM-QM9和GEOM-Drugs数据集上，ENBP实现98.9%构象覆盖率，误差0.090 Å，延迟低于1秒，速度超过扩散基线100倍以上且精度更高。在多体机器人推理中，传统环状信念传播在15个以上智能体时发散，而ENBP收敛且碰撞率接近零，等变误差达机器精度（~10⁻⁷），显著优于增强基线（10⁻¹）。
原文摘要 · Abstract (English)
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).
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