Estimatable variation neural networks and their application to ODEs and scalar hyperbolic conservation laws
<|reference_start|>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.<|reference_end|>
@article{lukáčová-medviďová2024estimatable,
    title={Estimatable variation neural networks and their application to ODEs and
  scalar hyperbolic conservation laws},
    author={M'aria Luk'av{c}ov'a-Medviv{d}ov'a, Simon Schneider},
    journal={arXiv preprint arXiv:2409.08909},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.08909},
    primaryClass={math.NA cs.NA}
}