  In this effort we propose a novel approach for reconstructing multivariate
functions from training data, by identifying both a suitable network
architecture and an initialization using polynomial-based approximations.
Training deep neural networks using gradient descent can be interpreted as
moving the set of network parameters along the loss landscape in order to
minimize the loss functional. The initialization of parameters is important for
iterative training methods based on descent. Our procedure produces a network
whose initial state is a polynomial representation of the training data. The
major advantage of this technique is from this initialized state the network
may be improved using standard training procedures. Since the network already
approximates the data, training is more likely to produce a set of parameters
associated with a desirable local minimum. We provide the details of the theory
necessary for constructing such networks and also consider several numerical
examples that reveal our approach ultimately produces networks which can be
effectively trained from our initialized state to achieve an improved
approximation for a large class of target functions.
