Learning to Discretize: Solving 1D Scalar Conservation Laws via Deep Reinforcement Learning (Communications in Computational Physics, 2020)
Conservation laws are considered to be fundamental laws of nature. It hasbroad applications in many fields, including physics, chemistry, biology, geology, andengineering. Solving the differential equations associated with conservation laws is amajor branch in computational mathematics. The recent success of machine learning,especially deep learning in areas such as computer vision and natural language processing, has attracted a lot of attention from the community of computational mathematics and inspired many intriguing works in combining machine learning with traditional methods. In this paper, we are the first to view numerical PDE solvers as anMDP and to use (deep) RL to learn new solvers. As proof of concept, we focus on1-dimensional scalar conservation laws. We deploy the machinery of deep reinforcement learning to train a policy network that can decide on how the numerical solutions should be approximated in a sequential and spatial-temporal adaptive manner.We will show that the problem of solving conservation laws can be naturally viewedas a sequential decision-making process, and the numerical schemes learned in such away can easily enforce long-term accuracy. Furthermore, the learned policy networkis carefully designed to determine a good local discrete approximation based on thecurrent state of the solution, which essentially makes the proposed method a meta-learning approach. In other words, the proposed method is capable of learning how todiscretize for a given situation mimicking human experts. Finally, we will provide details on how the policy network is trained, how well it performs compared with somestate-of-the-art numerical solvers such as WENO schemes, and supervised learningbased approach L3D and PINN, and how well it generalizes.
Keywords: Conservation laws; deep reinforcement learning; finite difference approximation; WENO.