Title: RoeNets: Predicting Discontinuity of Hyperbolic Systems from Continuous Data. Abstract: Predicting future discontinuous phenomena that are unobservable from the training data sets has been a challenging problem for scientific machine learning. In this paper, we introduce a novel learning paradigm to predict the emergence and evolution of various kinds of discontinuities for hyperbolic dynamic systems based on smooth observation data. At the heart of our approach is a templaterizable and data-driven Riemann solver that functions as a strong inductive prior to tackle the potential discontinuities.
The key design of our templaterized Riemann approximator is inspired by the classical Roe solver (P. L. Roe, J. Comput. Phys., vol. 43, 1981), which served as a fundamental mathematical tool for simulating various hyperbolic systems in computational physics. 
By carefully designing the computing primitives, data flow, and incorporating a novel pseudoinverse processing module, we enable our data-driven predictor to inherently satisfy all the essential mathematical criteria of a Roe hyperbolic solver derived from the first principles and hence deliver accurate predictions of hyperbolic dynamics.