A Low Rank Neural Representation of Entropy Solutions

TL;DR

This paper introduces a novel low rank neural network method to represent entropy solutions of nonlinear scalar conservation laws, capturing complex shock dynamics with linear embedded evolution.

Contribution

It develops a low rank neural representation that generalizes characteristics, enabling efficient approximation of complex entropy solutions with linear embedded dynamics.

Findings

Neural representation can approximate any entropy solution.

The method retains linearity of embedded dynamics.

Effective for complex shock topologies.

Abstract

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.