从模板预测轨迹的机器
Machines that Predict Trajectories from Templates
AI总结：
研究从存储模板库预测轨迹，无需识别生成模型。通过动态系统生成的轨迹库定义行为空间作预测机制，分析线性系统精确预测及相关特性，扩展到非线性系统，提供超越存储轨迹的基于模板的预测机器理论。
AI中文摘要：
我们研究从存储的输出模板库进行轨迹预测。给定未知轨迹的过去，目标是在不识别生成它的状态空间模型的情况下预测其未来。我们表明，由一个或多个动态系统生成的轨迹库定义了可作为预测机制的行为空间。对于线性系统，我们根据延续映射、行为包含和输出可见特征值的谱条件来表征精确预测。我们还分析了对噪声观测和噪声库的鲁棒性，推导了库外轨迹的误差界，并展示了互连约束如何将模板库组合成具有新兴模式的新行为空间。最后，我们将该框架扩展到输出轨迹包含或浸入有限维线性行为的非线性系统。这些结果提供了一种基于模板的预测机器理论，能够超越存储的轨迹进行泛化，在某些情况下，还能超越生成它们的系统。
英文摘要：
We study trajectory prediction from libraries of stored output templates. Given the past of an unknown trajectory, the goal is to predict its future without identifying the state-space model that generated it. We show that libraries of trajectories generated by one or more dynamical systems define behavioral spaces that can be used as prediction mechanisms. For linear systems, we characterize exact prediction in terms of continuation maps, behavioral containment, and spectral conditions on output-visible eigenvalues. We also analyze robustness to noisy observations and noisy libraries, derive error bounds for out-of-library trajectories, and show how interconnection constraints can compose template libraries into new behavioral spaces with emergent modes. Finally, we extend the framework to nonlinear systems whose output trajectories are contained in, or immersed into, finite-dimensional linear behaviors. These results provide a theory of template-based prediction machines capable of generalizing beyond the stored trajectories and, in some cases, beyond the systems that generated them.
