The problem of function approximation by neural dynamical systems has typically been approached in a top-down manner: Any continuous function can be approximated to an arbitrary accuracy by a sufficiently complex model with a given architecture. This can lead to high-complexity controls which are impractical in applications. In this paper, we take the opposite, constructive approach: We impose various structural restrictions on system dynamics and consequently characterize the class of functions that can be realized by such a system. The systems are implemented as a cascade interconnection of a neural stochastic differential equation (Neural SDE), a deterministic dynamical system, and a readout map. Both probabilistic and geometric (Lie-theoretic) methods are used to characterize the classes of functions realized by such systems.

翻译：神经动力系统函数逼近问题通常采用自上而下的方法处理：任何连续函数都可以通过给定架构下足够复杂的模型以任意精度逼近。这可能导致控制复杂度高，在实际应用中不切实际。本文采取相反的构造性方法：对系统动力学施加各类结构性限制，进而刻画此类系统所能实现的函数类。系统实现为神经随机微分方程（Neural SDE）、确定性动力系统与读出映射的级联互连结构。采用概率论与几何（李理论）方法共同刻画此类系统所能实现的函数类。