Estimatable variation neural networks and their application to ODEs and scalar hyperbolic conservation laws

We introduce estimatable variation neural networks (EVNNs), a class of neural networks that allow a computationally cheap estimate on the $BV$ norm motivated by the space $BMV$ of functions with bounded M-variation. We prove a universal approximation theorem for EVNNs and discuss possible implementations. We construct sequences of loss functionals for ODEs and scalar hyperbolic conservation laws for which a vanishing loss leads to convergence. Moreover, we show the existence of sequences of loss minimizing neural networks if the solution is an element of $BMV$. Several numerical test cases illustrate that it is possible to use standard techniques to minimize these loss functionals for EVNNs.