需求诱导的预测几何：动力系统有限资源预测
Requirement-Induced Predictive Geometry for Finite-Resource Prediction in Dynamical Systems
- SOBIN Institute LLC（SOBIN研究所有限公司）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对非线性动力系统预测中表示几何影响预测价值的问题，提出需求诱导预测几何，证明最优协方差具有逆预测形状，并给出预测增益的谱定律与上限，为自适应传感和状态估计提供基础。
AI中文摘要：
在非线性预测中，具有相同局部不确定性体积的两种状态表示可能具有截然不同的预测价值。我们考虑可微的有限时间动力学以及一个二次终端需求，该需求指定哪些终端状态差异是重要的。将此需求通过切映射拉回，定义了当前切空间上的需求诱导预测几何。该几何由动力学和需求固定；在此处使用的固定局部不确定性体积约束下，容量参数仅固定实现该几何的表示的尺度。我们证明，唯一最优协方差具有逆预测形状，并且等价地，最优性以终端需求几何中传播不确定性的各向同性化为特征。相对于基线各向同性表示的增益是预测谱的算术-几何平均比，使其成为预测各向异性的坐标不变度量。在欧几里得坐标中，这产生了一个通用的有限时间谱定律，对于二维保面积动力学，预测增益与最优几何的条件数之间存在精确恒等式。因此，更强的方向拉伸仅在伴随实现该拉伸所需的各向异性时增加潜在预测增益；限制该各向异性则施加了有限的增益上限。标准映射和双摆计算证明了这些关系在非线性哈密顿系统中的应用。该框架为自适应传感、状态估计和模型预测计算提供了基础，其中表示几何与预测任务后果相匹配。
英文摘要：
In nonlinear prediction, two state representations with the same local uncertainty volume can have radically different predictive value. We consider differentiable finite-time dynamics together with a quadratic terminal requirement that specifies which terminal state differences matter. Pulling this requirement back through the tangent map defines a requirement-induced predictive geometry on the present tangent space. This geometry is fixed by the dynamics and the requirement; under the fixed local uncertainty-volume constraint used here, the capacity parameter fixes only the scale of the representation that realizes it. We show that the unique optimal covariance has the inverse predictive shape and, equivalently, that optimality is characterized by isotropization of propagated uncertainty in the terminal requirement geometry. The gain over a baseline-isotropic representation is the arithmetic-to-geometric mean ratio of the predictive spectrum, making it a coordinate-invariant measure of predictive anisotropy. In Euclidean coordinates this yields a general finite-time spectral law, and for two-dimensional area-preserving dynamics an exact identity between predictive gain and the condition number of the optimal geometry. Stronger directional stretching therefore increases potential predictive gain only together with the anisotropy required to realize it; bounding that anisotropy imposes a finite gain ceiling. Standard-map and double-pendulum calculations demonstrate these relations in nonlinear Hamiltonian systems. The framework provides a basis for adaptive sensing, state estimation, and model-predictive computation in which representation geometry is matched to predicted task consequences.
