  We introduce Roe Neural Networks (RoeNets) that can predict the discontinuity
of the hyperbolic conservation laws (HCLs) based on short-term discontinuous
and even continuous training data. Our methodology is inspired by Roe
approximate Riemann solver (P. L. Roe, J. Comput. Phys., vol. 43, 1981, pp.
357--372), which is one of the most fundamental HCLs numerical solvers. In
order to accurately solve the HCLs, Roe argues the need to construct a Roe
matrix that fulfills "Property U", including diagonalizable with real
eigenvalues, consistent with the exact Jacobian, and preserving conserved
quantities. However, the construction of such matrix cannot be achieved by any
general numerical method. Our model made a breakthrough improvement in solving
the HCLs by applying Roe solver under a neural network perspective. To enhance
the expressiveness of our model, we incorporate pseudoinverses into a novel
context to enable a hidden dimension so that we are flexible with the number of
parameters. The ability of our model to predict long-term discontinuity from a
short window of continuous training data is in general considered impossible
using traditional machine learning approaches. We demonstrate that our model
can generate highly accurate predictions of evolution of convection without
dissipation and the discontinuity of hyperbolic systems from smooth training
data.
