RoeNets: Predicting Discontinuity of Hyperbolic Systems from Continuous Data

TL;DR

RoeNets are a novel neural network approach inspired by Roe's approximate Riemann solver, capable of predicting discontinuities in hyperbolic conservation laws from short-term continuous data, surpassing traditional methods.

Contribution

The paper introduces RoeNets, a neural network model that incorporates Roe's solver principles and pseudoinverses to predict hyperbolic system discontinuities from limited data, a task previously considered infeasible.

Findings

Accurately predicts discontinuities from short-term data

Generates high-fidelity evolution of convection without dissipation

Outperforms traditional machine learning approaches in long-term predictions

Abstract

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.