Multi-Agent Learning of Numerical Methods for Hyperbolic PDEs with Factored Dec-MDP
This provides a multi-agent learning approach for hyperbolic PDEs, which is incremental as it adapts existing frameworks to a new domain.
The paper tackles the problem of learning numerical methods for hyperbolic PDEs by formalizing it as a factored Dec-MDP, showing that different reward formulations lead to RL or behavior cloning, and achieving a homogeneous policy for agents that generalizes across spatial discretizations, episode lengths, dimensions, and equation types.
Factored decentralized Markov decision process (Dec-MDP) is a framework for modeling sequential decision making problems in multi-agent systems. In this paper, we formalize the learning of numerical methods for hyperbolic partial differential equations (PDEs), specifically the Weighted Essentially Non-Oscillatory (WENO) scheme, as a factored Dec-MDP problem. We show that different reward formulations lead to either reinforcement learning (RL) or behavior cloning, and a homogeneous policy could be learned for all agents under the RL formulation with a policy gradient algorithm. Because the trained agents only act on their local observations, the multi-agent system can be used as a general numerical method for hyperbolic PDEs and generalize to different spatial discretizations, episode lengths, dimensions, and even equation types.