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：基于有限元的高阶守恒律求解器具有高精度，但在间断附近因吉布斯现象面临挑战。基于物理洞察的人工黏性是该问题常用且有效的解决方案。本研究提出一种物理信息机器学习算法，以无监督范式自动化发现人工黏性模型。该算法受强化学习启发，通过自动微分构建以参考解差异定义的损失函数，逐单元训练神经网络（即黏性模型），实现了无需数据集的训练流程。我们将该算法集成至先进的龙格-库塔间断伽辽金求解器中验证其有效性，在一维和二维标量与向量问题（如Burgers方程和Euler方程）上进行了多组数值测试。结果表明，所提方法训练出的模型性能优于经典黏性模型。此外，研究发现习得的人工黏性模型具备跨问题和跨参数泛化能力。