Applied Mathematics Seminar——Conservative cell-average-based neural network method for nonlinear conservation laws
报告人： Jue Yan（Iowa State University）
时间：2026-06-30 10:30-11:30
地点：智华楼-四元厅-225
Abstract:
In this talk, we present the conservative version cell-average-based neural network (CANN) for nonlinear conservation laws under the one-dimensional setting. A simple feedforward network is explored to approximate the flux at the cell interface. Training is to identify the weights and biases which serve as the scheme coefficients of a finite volume type explicit one-step solver. The conservative CANN method can effectively learn a Riemann solver for the flux that resembles methods such as the Lax-Friedrich scheme, enabling it to capture entropy solutions involving shocks and rarefaction waves accurately. The method maintains the ability to take larger timesteps than traditional explicit solvers. For Burgers' equation, on both uniform and non-uniform meshes, we find that the network can accurately solve any initial value problem bounded by the training data. The method, trained solely on smooth solution data, can sharply resolve discontinuous solutions such as shocks. For the Euler equations, the conservative CANN method is feature-focused. Benchmark problems of the Sod shock tube, a simplified version of the Shu-Osher, and the double rarefaction problems are studied. The conservative CANN method is verified to capture entropy solutions to these problems consisting of rarefaction waves, shocks, contact discontinuities, along with their interactions.
Short Bio:
Dr. Yan is an associate professor of Mathematics at the Iowa State University. Her research focuses on the development, analysis and implementation of novel numerical methods for PDEs arising in physics and engineering applications. Specifically, her interest is on discontinuous Galerkin finite element method for gas dynamics applications. Her recent work includes the newly developed cell-average-based fast neural network solver for time-dependent problems. Her research has been continuously supported by the U.S. National Science Foundation and Simons Foundation. Dr. Yan received her Ph.D from Brown University in 2002 and spent four years as a postdoc at UCLA before joining Iowa State University.
