Colloquium with Jue Yan (2026-04-01)

"Conservative cell-average-based neural network method for nonlinear conservation laws"

Jue Yan
Department of Mathematics
Iowa State University

Wednesday, Apr 1, 2026

Colloquium - 499 DSL Seminar Room: 03:30 to 04:30 PM Eastern Time (US and Canada)

In-person attendance is requested.

499 DSL Seminar Room

Zoom access is intended for external (non-departmental) participants only.

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Meeting # 942 7359 5552

Zoom Meeting # 942 7359 5552

Abstract:

In this talk, we present the recently developed Cell-Average-Based Neural Network (CANN) for time-dependent PDEs. Spatial and temporal discretization in conventional methods is replaced by training a simple feedforward network to obtain an explicit one-step finite volume method. The well-trained network parameters function as the scheme coefficients. The network method has a minimum number of unknowns, which can be quickly trained on a small training dataset. Unlike conventional numerical methods, the CANN approach is not limited by the CFL conditions, enabling the use of significantly larger time steps. This leads to a highly efficient computational method for solving PDEs. The conservative version of the CANN method for nonlinear conservation laws will be discussed. The method, trained on the smooth-solution data from ONE initial-value problem, is verified to solve any initial-value problem involving shocks and rarefaction waves. The network can accurately learn a numerical flux, such as the Lax-Friedrichs flux, thereby guaranteeing convergence to the physically relevant entropy solutions. The extension of the method to non-uniform meshes will be considered. A bound-preserving version of the network method is presented for linear Advection equations.

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