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Title: Learning to Discretize: Solving 1D Scalar Conservation Laws via Deep Reinforcement Learning

Paper content:
Conservation laws are considered to be fundamental laws of nature.
It has broad application in many fields including physics, chemistry, biology, geology, and engineering.
Solving the differential equations associated with conservation laws is a major branch in computational mathematics.
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 explore the possibility and benefit of solving nonlinear conservation laws using deep reinforcement learning.
As a proof of concept, we focus on 1-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 viewed as a sequential decision making process and the numerical schemes learned in such a way can easily enforce long-term accuracy.
Furthermore, the learned policy network is carefully designed to determine a good local discrete approximation based on the current 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 to discretize 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 some state-of-the-art numerical solvers such as WENO schemes, and how well it generalizes.
Our code is released anomynously at \url{https://github.com/qwerlanksdf/L2D}.
Conservation laws are considered to be one of the fundamental laws of nature, and has broad applications in multiple fields such as physics, chemistry, biology, geology, and engineering.
For example, Burger's equation, a very classic partial differential equation (PDE) in conservation laws, has important applications in fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow.
Solving the differential equations associated with conservation laws has been a major branch of computational mathematics (LeVeque, 1992; 2002) , and a lot of effective methods have been proposed, from classic methods such as the upwind scheme, the Lax-Friedrichs scheme, to the advanced ones such as the ENO/WENO schemes (Liu et al., 1994; Shu, 1998) , the flux-limiter methods (Jerez Galiano & Uh Zapata, 2010) , and etc.
In the past few decades, these traditional methods have been proven successful in solving conservation laws.
Nonetheless, the design of some of the high-end methods heavily relies on expert knowledge and the coding of these methods can be a laborious process.
To ease the usage and potentially improve these traditional algorithms, machine learning, especially deep learning, has been recently incorporated into this field.
For example, the ENO scheme requires lots of 'if/else' logical judgments when used to solve complicated system of equations or high-dimensional equations.
This very much resembles the old-fashioned expert systems.
The recent trend in artificial intelligence (AI) is to replace the expert systems by the so-called 'connectionism', e.g., deep neural networks, which leads to the recent bloom of AI.
Therefore, it is natural and potentially beneficial to introduce deep learning in traditional numerical solvers of conservation laws.
In this paper, we proposed a general framework to learn how to solve 1-dimensional conservation laws via deep reinforcement learning.
We first discussed how the procedure of numerically solving conservation laws can be naturally cast in the form of Markov Decision Process.
We then elaborated how to relate notions in numerical schemes of PDEs with those of reinforcement learning.
In particular, we introduced a numerical flux policy which was able to decide on how numerical flux should be designed locally based on the current state of the solution.
We carefully design the action of our RL policy to make it a meta-learner.
Our numerical experiments showed that the proposed RL based solver was able to outperform high order WENO and was well generalized in various cases.
As part of the future works, we would like to consider using the numerical flux policy to inference more complicated numerical fluxes with guaranteed consistency and stability.
Furthermore, we can use the proposed framework to learn a policy that can generate adaptive grids and the associated numerical schemes.
Lastly, we would like consider system of conservation laws in 2nd and 3rd dimensional space.
A COMPLEMENTARY EXPERIMENTS

assistant: text: We observe that numerical PDE solvers can be regarded as Markov Desicion Processes, and propose to use Reinforcement Learning to solve 1D scalar Conservation Laws