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We introduce Roe Neural Networks (RoeNets) that can predict the discontinuityof the hyperbolic conservation laws (HCLs) based on short-term discontinuousand even continuous training data. Our methodology is inspired by Roeapproximate Riemann solver (P. L. Roe, J. Comput. Phys., vol. 43, 1981, pp.357--372), which is one of the most fundamental HCLs numerical solvers. Inorder to accurately solve the HCLs, Roe argues the need to construct a Roematrix that fulfills "Property U", including diagonalizable with realeigenvalues, consistent with the exact Jacobian, and preserving conservedquantities. However, the construction of such matrix cannot be achieved by anygeneral numerical method. Our model made a breakthrough improvement in solvingthe HCLs by applying Roe solver under a neural network perspective. To enhancethe expressiveness of our model, we incorporate pseudoinverses into a novelcontext to enable a hidden dimension so that we are flexible with the number ofparameters. The ability of our model to predict long-term discontinuity from ashort window of continuous training data is in general considered impossibleusing traditional machine learning approaches. We demonstrate that our modelcan generate highly accurate predictions of evolution of convection withoutdissipation and the discontinuity of hyperbolic systems from smooth trainingdata.

RoeNets: Predicting Discontinuity of Hyperbolic Systems from Continuous Data