Finite element discretizations of problems in computational physics often rely on adaptive mesh refinement (AMR) to preferentially resolve regions containing important features during simulation. However, these spatial refinement strategies are often heuristic and rely on domain-specific knowledge or trial-and-error. We treat the process of adaptive mesh refinement as a local, sequential decision-making problem under incomplete information, formulating AMR as a partially observable Markov decision process. Using a deep reinforcement learning approach, we train policy networks for AMR strategy directly from numerical simulation. The training process does not require an exact solution or a high-fidelity ground truth to the partial differential equation at hand, nor does it require a pre-computed training dataset. The local nature of our reinforcement learning formulation allows the policy network to be trained inexpensively on much smaller problems than those on which they are deployed. The methodology is not specific to any particular partial differential equation, problem dimension, or numerical discretization, and can flexibly incorporate diverse problem physics. To that end, we apply the approach to a diverse set of partial differential equations, using a variety of high-order discontinuous Galerkin and hybridizable discontinuous Galerkin finite element discretizations. We show that the resultant deep reinforcement learning policies are competitive with common AMR heuristics, generalize well across problem classes, and strike a favorable balance between accuracy and cost such that they often lead to a higher accuracy per problem degree of freedom.
翻译:计算物理学中的问题的局部分解性要素往往依赖于适应性网格改进(AMR),以优先解决模拟中含有重要特征的区域。然而,这些空间改进战略往往是超常的,依赖特定领域的知识或试验和试验。我们把适应性网格改进过程视为信息不完整的局部、顺序决策问题,将AMR发展为部分可观察的马尔科夫决策过程。我们采用深度强化学习方法,直接从数字模拟中为AMR战略培训政策网络。培训进程不需要精确的解决方案或高异性地面真象来取代手头部分差异方程式,也不需要预先计算的培训数据集。我们强化学习的当地性质使得政策网络能够以费用低廉的方式在远小于其部署时的问题上接受培训。这种方法并不具体针对任何特定的局部差异方程式、问题层面或数字离散化,并且可以灵活地纳入多种问题物理学。为此,我们将这一方法应用于一系列不同的部分差异方程式,使用各种高等级的不连续的准确性、不精确性、不连续性、不连续性、不连续性、不连续性、不连续性、不固定性、不固定性、不固定性、不固定性、不固定性、不固定性、不固定性、不固定性、不固定性、不固定性、不固定性的政策、不固定性、不固定性、不固定性、不固定性、不固定性、不固定性、不固定性、不固定性、不固定性、不固定性、不固定性等等等。