  The training of artificial neural networks (ANNs) with rectified linear unit
(ReLU) activation via gradient descent (GD) type optimization schemes is
nowadays a common industrially relevant procedure. Till this day in the
scientific literature there is in general no mathematical convergence analysis
which explains the numerical success of GD type optimization schemes in the
training of ANNs with ReLU activation. GD type optimization schemes can be
regarded as temporal discretization methods for the gradient flow (GF)
differential equations associated to the considered optimization problem and,
in view of this, it seems to be a natural direction of research to first aim to
develop a mathematical convergence theory for time-continuous GF differential
equations and, thereafter, to aim to extend such a time-continuous convergence
theory to implementable time-discrete GD type optimization methods. In this
article we establish two basic results for GF differential equations in the
training of fully-connected feedforward ANNs with one hidden layer and ReLU
activation. In the first main result of this article we establish in the
training of such ANNs under the assumption that the probability distribution of
the input data of the considered supervised learning problem is absolutely
continuous with a bounded density function that every GF differential equation
admits for every initial value a solution which is also unique among a suitable
class of solutions. In the second main result of this article we prove in the
training of such ANNs under the assumption that the target function and the
density function of the probability distribution of the input data are
piecewise polynomial that every non-divergent GF trajectory converges with an
appropriate rate of convergence to a critical point and that the risk of the
non-divergent GF trajectory converges with rate 1 to the risk of the critical
point.
