论文标题
通过伴随方程从其轨迹学习微分方程的参数
Learning the parameters of a differential equation from its trajectory via the adjoint equation
论文作者
论文摘要
本文有助于加强机器学习与微分方程理论之间的关系。在这种情况下，拟合参数的逆问题，而微分方程的初始条件与某些测量值构成了关键问题。本文探讨了一个可以用于构建损失功能家族的抽象，目的是将初始值问题解决方案拟合到一组离散或连续测量中。据表明，伴随方程的扩展可以用来推导损失函数的梯度，作为机器学习中反向传播的连续类似物。提供了数值证据，表明在合理控制的情况下，获得的梯度可以在梯度下降中使用，以将初始值问题解决方案拟合到一组连续的嘈杂测量值中，以及一组在不确定时间记录的离散噪声测量值。
The paper contributes to strengthening the relation between machine learning and the theory of differential equations. In this context, the inverse problem of fitting the parameters, and the initial condition of a differential equation to some measurements constitutes a key issue. The paper explores an abstraction that can be used to construct a family of loss functions with the aim of fitting the solution of an initial value problem to a set of discrete or continuous measurements. It is shown, that an extension of the adjoint equation can be used to derive the gradient of the loss function as a continuous analogue of backpropagation in machine learning. Numerical evidence is presented that under reasonably controlled circumstances the gradients obtained this way can be used in a gradient descent to fit the solution of an initial value problem to a set of continuous noisy measurements, and a set of discrete noisy measurements that are recorded at uncertain times.
