A Low Rank Neural Representation of Entropy Solutions
This work addresses the problem of efficiently representing complex shock topologies in conservation laws for researchers in computational mathematics and physics, though it appears incremental as it builds on existing methods like characteristics and neural representations.
The authors tackled the representation of entropy solutions for nonlinear scalar conservation laws by constructing a new compositional representation that generalizes the method of characteristics, and they discretized it as a low-rank implicit neural representation that can approximate any entropy solution with a fixed number of layers and a small number of coefficients.
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.