  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.
