  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.
