论文标题
颂歌流的路径不同
Path differentiability of ODE flows
论文作者
论文摘要
我们考虑路径可区分矢量场驱动的普通微分方程（OD）的流动。路径可区分函数构成了Lipschitz函数的适当子类，该函数接纳了保守的梯度，这是一般性衍生物与基本演算规则兼容的概念。我们的主要结果指出，这种流程继承了驱动向量场的路径可不同性能。我们确实表明，灵敏度差异包含物给出的衍生物的正向传播为流动提供了保守的雅各布式。这允许提出一个非平滑版本的伴随方法，可以将其应用于ODE约束下的积分成本。该结果构成了小型步骤一阶方法的应用理论基础，以解决与参数化约束的广泛的非平滑优化问题。基于提议的非平滑伴随的小步骤一阶方法的收敛来说明这一点。
We consider flows of ordinary differential equations (ODEs) driven by path differentiable vector fields. Path differentiable functions constitute a proper subclass of Lipschitz functions which admit conservative gradients, a notion of generalized derivative compatible with basic calculus rules. Our main result states that such flows inherit the path differentiability property of the driving vector field. We show indeed that forward propagation of derivatives given by the sensitivity differential inclusions provide a conservative Jacobian for the flow. This allows to propose a nonsmooth version of the adjoint method, which can be applied to integral costs under an ODE constraint. This result constitutes a theoretical ground to the application of small step first order methods to solve a broad class of nonsmooth optimization problems with parametrized ODE constraints. This is illustrated with the convergence of small step first order methods based on the proposed nonsmooth adjoint.
