AI 中文总结
研究由神经动力学控制的系统中微分方程的精确整体可控性，基于神经插值问题提出构造性解决方案，表明深度为二的神经网络情况下插值问题可化为线性方程组求解。
AI 中文摘要
我们研究了一个由神经动力学控制的系统。神经动力学是具有大量层的深度神经网络的模型。对于右侧由神经网络给出的微分方程，我们分析了该系统的精确整体可控性。此属性在机器学习中起着至关重要的作用。它涉及用单个神经动力学同时学习执行不同任务的能力：精确整体可控性要求用一组控制参数将\(N\)个不同的初始状态引导到相应的\(N\)个目标状态。我们提出了该问题的一个构造性解决方案。该构造基于神经插值问题的解。我们表明，如果微分方程的右侧由深度为二的神经网络给出，插值问题可简化为线性方程组的解。
英文摘要
We study a system that is governed by neural dynamics. Neural dynamics are a model for deep neural networks with a large number of layers. For a differential equation where the right-hand side is given by a neural network, we analyze the exact ensemble controllability of the system. This property plays an essential role in Machine Learning. It concerns the ability to learn to perform different tasks simultaneously with a single neural dynamics: Exact ensemble controllability requires that $N$ different initial states are steered to corresponding $N$ target states with a single set of control parameters. We present a constructive solution to the problem. The construction is based on the solution of a neural interpolation problem. We show that if the right-hand side of the differential equation is given by a neural network of depth two, the interpolation problem can be reduced to the solution of a system of linear equations.
