Large-scale finite element simulations of complex physical systems governed by partial differential equations crucially depend on adaptive mesh refinement (AMR) to allocate computational budget to regions where higher resolution is required. Existing scalable AMR methods make heuristic refinement decisions based on instantaneous error estimation and thus do not aim for long-term optimality over an entire simulation. We propose a novel formulation of AMR as a Markov decision process and apply deep reinforcement learning (RL) to train refinement policies directly from simulation. AMR poses a new problem for RL in that both the state dimension and available action set changes at every step, which we solve by proposing new policy architectures with differing generality and inductive bias. The model sizes of these policy architectures are independent of the mesh size and hence scale to arbitrarily large and complex simulations. We demonstrate in comprehensive experiments on static function estimation and the advection of different fields that RL policies can be competitive with a widely-used error estimator and generalize to larger, more complex, and unseen test problems.

本研究提出一种新的有限元自适应网格细化方法，将其视为马尔可夫决策过程，并应用深度强化学习进行训练，从而解决了传统方法对瞬时误差估计的依赖问题，并通过静态函数估计和时间依赖方程等实验验证了该方法的可行性和有效性。

自适应网格细化的强化学习