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

TL;DR

This paper introduces estimatable variation neural networks (EVNNs) that efficiently estimate the BV norm, with theoretical guarantees and practical applications to scalar hyperbolic conservation laws, demonstrating their effectiveness through numerical tests.

Contribution

The paper proposes EVNNs, a new neural network class that enables cheap BV norm estimation and proves their universal approximation capabilities for functions with bounded M-variation.

Findings

EVNNs can accurately estimate the BV norm.

Sequences of loss functionals lead to convergence for hyperbolic conservation laws.

Numerical tests confirm the practical effectiveness of EVNNs.

Abstract

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.