A Low Rank Neural Representation of Entropy Solutions
<|reference_start|>A Low Rank Neural Representation of Entropy Solutions: We construct a new representation of entropy solutions to nonlinear scalar conservation laws with a smooth convex flux function in a single spatial dimension. The representation is a generalization of the method of characteristics and posseses a compositional form. While it is a nonlinear representation, the embedded dynamics of the solution in the time variable is linear. This representation is then discretized as a manifold of implicit neural representations where the feedforward neural network architecture has a low rank structure. Finally, we show that the low rank neural representation with a fixed number of layers and a small number of coefficients can approximate any entropy solution regardless of the complexity of the shock topology, while retaining the linearity of the embedded dynamics.<|reference_end|>
@article{rim2024a,
    title={A Low Rank Neural Representation of Entropy Solutions},
    author={Donsub Rim, Gerrit Welper},
    journal={arXiv preprint arXiv:2406.05694},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.05694},
    primaryClass={math.NA cs.LG cs.NA}
}