We propose a data-driven physics-informed finite-volume scheme for the approximation of small-scale dependent shocks. Nonlinear hyperbolic conservation laws with non-convex fluxes allow nonclassical shock wave solutions. In this work, we consider the cubic scalar conservation law as representative of such systems. As standard numerical schemes fail to approximate nonclassical shocks, schemes with controlled dissipation and schemes with well-controlled dissipation have been introduced by LeFloch and Mohammadian and by Ernest and coworkers, respectively. Emphasis has been placed on matching the truncation error of the numerical scheme with physically relevant small-scale mechanisms. However, aforementioned schemes can introduce oscillations as well as excessive dissipation around shocks. In our approach, a convolutional neural network is used for an adaptive nonlinear flux reconstruction. Based on the local flow field, the network combines local interpolation polynomials with a regularization term to form the numerical flux. This allows to modify the discretization error by nonlinear terms. In a supervised learning task, the model is trained to predict the time evolution of exact solutions to Riemann problems. The model is physics-informed as it respects the underlying conservation law. Numerical experiments for the cubic scalar conservation law show that the resulting method is able to approximate nonclassical shocks very well. The adaptive reconstruction suppresses oscillations and enables sharp shock capturing. Generalization to unseen shock configurations, smooth initial value problems, and shock interactions is robust and shows very good results.

我们提出了一种数据驱动的，基于物理学的有限​​体积方案，用于近似小规模的相依冲击。具有非凸通量的非线性双曲守恒律允许非经典冲击波解。在这项工作中，我们认为立方标量守恒定律是此类系统的代表。由于标准数值方案无法近似非经典冲击，因此LeFloch和Mohammadian以及Ernest和同事分别引入了具有受控耗散的方案和具有良好受控耗散的方案。重点放在将数值方案的截断误差与物理上相关的小规模机制相匹配。但是，上述方案可能会在振荡周围引入振荡以及过度耗散。在我们的方法中，卷积神经网络用于自适应非线性通量重构。基于局部流场，网络将局部插值多项式与正则项组合在一起以形成数值通量。这允许通过非线性项来修改离散化误差。在有监督的学习任务中，训练模型以预测黎曼问题的精确解的时间演变。该模型是符合物理原理的，因为它遵守了基本的守恒定律。三次标量守恒律的数值实验表明，所得到的方法能够很好地逼近非经典冲击。自适应重建可抑制振荡，并能捕捉到强烈的震动。推广到看不见的冲击配置，平滑初始值问题，