让模型保持对称性，训练后仍能零样本泛化到所有方向。
Exact equivariance, kept through training, buys zero-shot generalisation across the symmetry group
- 用等变编码器和预测器构建潜空间模型，保证损失函数严格对称。
- 训练后误差在旋转群上完全平坦，3D任务误差仅为非等变基线的1/17.2。
- 适合追求可证明泛化能力的机器学习研究者与机器人控制开发者。
基于等变编码器和预测器构建的潜空间模型，继承了训练损失的精确对称性：当动力学在潜变量上由正交表示ρ(g)作用于群G时，一步预测相对均方误差（relMSE）在整个群上严格不变，因此只需拟合一个方向就能确定整个轨道上的表现。该对称性在真实训练中得以保留——无论使用Muon/AdamW+EMA+VICReg优化器，残差均小于10⁻⁶。一步误差在群上保持平坦（5次种子中位数：等变模型×1.00，高容量非等变基线×12.7（2D），×17.2（3D）），而后者仅在分布内拟合良好，无法泛化至分布外。真实机器人DROID末端执行器轨迹验证：等变模型在轨道上误差恒定（×1.000，旋转残差1.5×10⁻¹⁶），基线模型误差扩大×11，且规模大4.5倍。注意：平坦性是必要条件而非充分条件，它不降低内部误差水平（3D relMSE≈0.43），仅保证跨群误差一致。相同对称性可推广至闭环控制：匹配的等变规划器使控制误差在群上不变（2D/SO(2)为浮点精确，3D/SE(3)统计平坦）。对比增强、扩展规模、软等变性等策略均无法达到此精度。这是认证世界模型计划（arXiv:2606.13092, 2606.24945, 2606.24946）的基础：平坦性传递能力，信任边界由此生成。
原文摘要 · Abstract (English)
A latent world model built from an equivariant encoder and predictor inherits a provable symmetry of its training loss: when the dynamics carries a group $G$ acting on latents by an orthogonal representation $ρ(g)$, the one-step prediction relMSE is exactly invariant across the whole group, so fitting a restricted slice of orientations mathematically determines it on the entire orbit. The symmetry survives a real Muon/AdamW+EMA+VICReg run -- composed residual $\sim 10^{-6}$ after training, under any optimiser (intrinsic Vector-Neuron/e3nn parametrisation) -- and one-step error is flat across the group (5-seed medians: equivariant $\times 1.00$ vs a higher-capacity non-equivariant baseline $\times 12.7$ in 2D, $\times 17.2$ in 3D), while that baseline fits the slice but breaks out-of-distribution. The flatness is not a synthetic artefact: on real-robot DROID end-effector trajectories the equivariant model stays flat across the orbit ($\times 1.000$, rotation residual $1.5\times 10^{-16}$) while a $4.5\times$-larger baseline degrades $\times 11$. One caution is load-bearing: flatness is necessary, not sufficient -- the theorem transports the in-distribution error level unchanged but does not lower it (3D relMSE $\approx 0.43$): across-group error is constant, not low. The same isometry lifts to a closed-loop corollary: under a matching equivariant planner the control error is invariant across the group -- float-floor-exact in 2D/SO(2), statistically flat in 3D/SE(3). Stress-tested against Sutton's Bitter Lesson (augmentation, scale, soft-equivariance), each closes at most the across-group task metric, never the float-floor exactness. This is the generalisation-side foundation of a certified-world-models programme (arXiv:2606.13092, 2606.24945, 2606.24946): flatness transports competence, and the trust bounds built on it are downstream products.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。