  We propose an approach to solving partial differential equations (PDEs) using
a set of neural networks which we call Neural Basis Functions (NBF). This NBF
framework is a novel variation of the POD DeepONet operator learning approach
where we regress a set of neural networks onto a reduced order Proper
Orthogonal Decomposition (POD) basis. These networks are then used in
combination with a branch network that ingests the parameters of the prescribed
PDE to compute a reduced order approximation to the PDE. This approach is
applied to the steady state Euler equations for high speed flow conditions
(mach 10-30) where we consider the 2D flow around a cylinder which develops a
shock condition. We then use the NBF predictions as initial conditions to a
high fidelity Computational Fluid Dynamics (CFD) solver (CFD++) to show faster
convergence. Lessons learned for training and implementing this algorithm will
be presented as well.
