  The inverse problem of supervised reconstruction of depth-variable
(time-dependent) parameters in a neural ordinary differential equation (NODE)
is considered, that means finding the weights of a residual network with time
continuous layers. The NODE is treated as an isolated entity describing the
full network as opposed to earlier research, which embedded it between pre- and
post-appended layers trained by conventional methods. The proposed parameter
reconstruction is done for a general first order differential equation by
minimizing a cost functional covering a variety of loss functions and penalty
terms. A nonlinear conjugate gradient method (NCG) is derived for the
minimization. Mathematical properties are stated for the differential equation
and the cost functional. The adjoint problem needed is derived together with a
sensitivity problem. The sensitivity problem can estimate changes in the
network output under perturbation of the trained parameters. To preserve
smoothness during the iterations the Sobolev gradient is calculated and
incorporated. As a proof-of-concept, numerical results are included for a NODE
and two synthetic datasets, and compared with standard gradient approaches (not
based on NODEs). The results show that the proposed method works well for deep
learning with infinite numbers of layers, and has built-in stability and
smoothness.
