论文标题
神经颂歌控制连续性方程的轨迹近似
Neural ODE Control for Trajectory Approximation of Continuity Equation
论文作者
论文摘要
我们考虑了对应于神经普通微分方程（ODE）的连续性方程的可控性问题，该方程描述了流动如何推动概率度量。我们表明，受控的连续性方程具有非常强大的可控性能。特别是，与界面载体矢量场相对应的连续性方程的给定解定义了概率度量集的轨迹。对于这种轨迹，我们表明神经颂的分段恒定训练权重使与神经极相对应的连续性方程的解已任意接近它。作为这种结果的推论，我们确定神经ODE的连续性方程在一组紧凑型概率指标上近似可控，这些概率度量与Lebesgue度量绝对连续。
We consider the controllability problem for the continuity equation, corresponding to neural ordinary differential equations (ODEs), which describes how a probability measure is pushedforward by the flow. We show that the controlled continuity equation has very strong controllability properties. Particularly, a given solution of the continuity equation corresponding to a bounded Lipschitz vector field defines a trajectory on the set of probability measures. For this trajectory, we show that there exist piecewise constant training weights for a neural ODE such that the solution of the continuity equation corresponding to the neural ODE is arbitrarily close to it. As a corollary to this result, we establish that the continuity equation of the neural ODE is approximately controllable on the set of compactly supported probability measures that are absolutely continuous with respect to the Lebesgue measure.
