Finite element-based high-order solvers of conservation laws offer large accuracy but face challenges near discontinuities due to the Gibbs phenomenon. Artificial viscosity is a popular and effective solution to this problem based on physical insight. In this work, we present a physics-informed machine learning algorithm to automate the discovery of artificial viscosity models in a non-supervised paradigm. The algorithm is inspired by reinforcement learning and trains a neural network acting cell-by-cell (the viscosity model) by minimizing a loss defined as the difference with respect to a reference solution thanks to automatic differentiation. This enables a dataset-free training procedure. We prove that the algorithm is effective by integrating it into a state-of-the-art Runge-Kutta discontinuous Galerkin solver. We showcase several numerical tests on scalar and vectorial problems, such as Burgers' and Euler's equations in one and two dimensions. Results demonstrate that the proposed approach trains a model that is able to outperform classical viscosity models. Moreover, we show that the learnt artificial viscosity model is able to generalize across different problems and parameters.

翻译：有限元高阶守恒律求解器虽能提供高精度，但因吉布斯现象在间断附近面临挑战。人工黏性是基于物理见解解决该问题的常用有效方法。本文提出一种物理信息机器学习算法，以无监督方式自动发现人工黏性模型。该算法受强化学习启发，通过自动微分最小化与参考解的差异损失，训练一个逐单元作用的神经网络（即黏性模型），从而实现无需数据集的训练过程。我们将该算法集成至先进龙格-库塔间断伽辽金求解器中，验证其有效性。通过一维和二维标量及矢量问题（如Burgers方程和Euler方程）的数值测试表明，所提方法训练的模型性能优于经典黏性模型。此外，该学习得到的人工黏性模型能够跨不同问题和参数进行泛化。