发表机构
Allen Institute（艾伦研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明有限维光滑自主系统无法精确稳定生成时间反转的混沌动力学，如洛伦兹吸引子，通过熵矛盾揭示了时间之箭的存在。
AI 中文摘要
稳定的自主生成是否会选择时间方向？这个问题既关乎学习模型自主生成规定动力学，也关乎设计控制器使不变集在不改变其动力学的情况下成为吸引集。周期运动和准周期环面旋转在任一方向上都允许稳定生成。然而，对于混沌吸引子，反转方程会重现其反向轨迹，同时将吸引变为排斥。额外的潜变量或自主反馈能否在不改变反向运动的情况下恢复吸引？我们证明了有限维自主系统（具有$C^2$向量场、紧致吸引集和连续读出）对这种精确生成存在障碍。该结果适用于经典洛伦兹吸引子，更一般地，适用于具有正度量熵遍历不变测度且其内在局部稳定纤维在正测度集上完全断开的动力学。证明将这一几何性质与熵矛盾联系起来：一个假设的吸引实现将迫使输出熵消失，与规定的混沌动力学相悖。我们用储层计算示例来说明时间反转混沌的稳定自主生成的困难：在数据驱动网络时，两个时间方向都被紧密跟踪，但只有正向示例在自主生成期间保留了洛伦兹几何。该定理识别了一类动力学，在满足所述假设的有限维光滑模型中，无论学习过程或潜变量数量如何，都无法精确稳定自主生成。
英文摘要
Does stable autonomous generation select a direction of time? This question matters both for learning models that autonomously generate prescribed dynamics and for designing controllers that make an invariant set attracting without changing its dynamics. Periodic motion and quasiperiodic torus rotations admit stable generation in either direction. For a chaotic attractor, however, reversing the equations reproduces its backward trajectories while turning attraction into repulsion. Can additional latent variables or autonomous feedback restore attraction without changing the reversed motion? We prove an obstruction to such exact generation by finite-dimensional autonomous systems with $C^2$ vector fields, compact attracting sets, and continuous readouts. The result applies to the classical Lorenz attractor and, more generally, to dynamics with an ergodic invariant measure of positive metric entropy whose intrinsic local stable fibers are totally disconnected on a set of positive measure. The proof connects this geometry to an entropy contradiction: a hypothetical attracting realization would force the output entropy to vanish, contrary to the prescribed chaotic dynamics. We use reservoir-computing examples to illustrate the difficulty of stable autonomous generation for time-reversed chaos: both time directions are closely tracked while the networks are driven by data, but only the forward example retains Lorenz geometry during autonomous generation. The theorem identifies a class of dynamics that cannot be learned for exact, stable autonomous generation by finite-dimensional smooth models satisfying the stated assumptions, regardless of the learning procedure or the number of latent variables.
