  We provide a numerical analysis and computation of neural network projected
schemes for approximating one dimensional Wasserstein gradient flows. We
approximate the Lagrangian mapping functions of gradient flows by the class of
two-layer neural network functions with ReLU (rectified linear unit) activation
functions. The numerical scheme is based on a projected gradient method, namely
the Wasserstein natural gradient, where the projection is constructed from the
$L^2$ mapping spaces onto the neural network parameterized mapping space. We
establish theoretical guarantees for the performance of the neural projected
dynamics. We derive a closed-form update for the scheme with well-posedness and
explicit consistency guarantee for a particular choice of network structure.
General truncation error analysis is also established on the basis of the
projective nature of the dynamics. Numerical examples, including gradient drift
Fokker-Planck equations, porous medium equations, and Keller-Segel models,
verify the accuracy and effectiveness of the proposed neural projected
algorithm.
