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AN ALTERNATING FLUX LEARNING METHOD FOR MULTIDIMENSIONAL NONLINEAR CONSERVATION LAWS
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DOI：10.1137/23M1556605.png)
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Abstract
En 中文
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 (*) u(t) + f(u)(x) + g(u)(y) = 0. Straightforward extension of the method from the 1D to the 2D problem results in poor identification of the unknown f(u) and g(u). 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 f and g in the general case when they are nonconvex and unequal.
Keywords:
scalar nonlinear conservation law
symbolic multilayer neural network
entropy consistent discrete numerical scheme
joint equations training
alternating equations training
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