An Alternating Flux Learning Method for Multidimensional Nonlinear Conservation Laws

SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page C421-C447, August 2024.
Abstract. In a recent work [Q. Li and S. Evje, Netw. Heterog. Media, 18 (2023), pp. 48–79], it was explored how to identify the unknown flux function in a one-dimensional scalar conservation law. Key ingredients are symbolic neural networks to represent the candidate flux functions, entropy-satisfying numerical schemes, and a proper combination of initial data. The purpose of this work is to extend this methodology to a two-dimensional scalar conservation law ([math]) [math]. Straightforward extension of the method from the 1D to the 2D problem results in poor identification of the unknown [math] and [math]. Relying on ideas from joint and alternating equations training, a learning strategy is designed that enables accurate identification of the flux functions, even when 2D observations are sparse. It involves an alternating flux training approach where a first set of candidate flux functions obtained from joint training is improved through an alternating direction-dependent training strategy. Numerical investigations demonstrate that the method can effectively identify the true underlying flux functions [math] and [math] in the general case when they are nonconvex and unequal.