  Motivated by the gap between theoretical optimal approximation rates of deep
neural networks (DNNs) and the accuracy realized in practice, we seek to
improve the training of DNNs. The adoption of an adaptive basis viewpoint of
DNNs leads to novel initializations and a hybrid least squares/gradient descent
optimizer. We provide analysis of these techniques and illustrate via numerical
examples dramatic increases in accuracy and convergence rate for benchmarks
characterizing scientific applications where DNNs are currently used, including
regression problems and physics-informed neural networks for the solution of
partial differential equations.
