  Using neural networks to solve variational problems, and other scientific
machine learning tasks, has been limited by a lack of consistency and an
inability to exactly integrate expressions involving neural network
architectures. We address these limitations by formulating a novel neural
network architecture that combines a polynomial mixture-of-experts model with
free knot B1-spline basis functions. Effectively, our architecture performs
piecewise polynomial approximation on each cell of a trainable partition of
unity. Our architecture exhibits both $h$- and $p$- refinement for regression
problems at the convergence rates expected from approximation theory, allowing
for consistency in solving variational problems. Moreover, this architecture,
its moments, and its partial derivatives can all be integrated exactly,
obviating a reliance on sampling or quadrature and enabling error-free
computation of variational forms. We demonstrate the success of our network on
a range of regression and variational problems that illustrate the consistency
and exact integrability of our network architecture.
