Reinforcement learning (RL) is a promising method to solve control problems. However, model-free RL algorithms are sample inefficient and require thousands if not millions of samples to learn optimal control policies. A major source of computational cost in RL corresponds to the transition function, which is dictated by the model dynamics. This is especially problematic when model dynamics is represented with coupled PDEs. In such cases, the transition function often involves solving a large-scale discretization of the said PDEs. We propose a multilevel RL framework in order to ease this cost by exploiting sublevel models that correspond to coarser scale discretization (i.e. multilevel models). This is done by formulating an approximate multilevel Monte Carlo estimate of the objective function of the policy and / or value network instead of Monte Carlo estimates, as done in the classical framework. As a demonstration of this framework, we present a multilevel version of the proximal policy optimization (PPO) algorithm. Here, the level refers to the grid fidelity of the chosen simulation-based environment. We provide two examples of simulation-based environments that employ stochastic PDEs that are solved using finite-volume discretization. For the case studies presented, we observed substantial computational savings using multilevel PPO compared to its classical counterpart.
Ershov Institute of Informatics Systems（埃尔绍夫信息系统研究所）
We propose a novel Reinforcement Learning model for discrete environments, which is inherently interpretable and supports the discovery of deep subgoal hierarchies. In the model, an agent learns information about environment in the form of probabilistic rules, while policies for (sub)goals are learned as combinations thereof. No reward function is required for learning; an agent only needs to be given a primary goal to achieve. Subgoals of a goal G from the hierarchy are computed as descriptions of states, which if previously achieved increase the total efficiency of the available policies for…
Peking University（北京大学）；Eastern Institute for Advanced Study（东方高等研究院）；Eastern Institute of Technology（东方理工大学）；National Center for Applied Mathematics Shenzhen（深圳国家应用数学中心）；Southern University of Science and Technology（南方科技大学）
The working mechanisms of complex natural systems tend to abide by concise and profound partial differential equations (PDEs). Methods that directly mine equations from data are called PDE discovery, which reveals consistent physical laws and facilitates our adaptive interaction with the natural world. In this paper, an enhanced deep reinforcement-learning framework is proposed to uncover symbolically concise open-form PDEs with little prior knowledge. Particularly, based on a symbol library of basic operators and operands, a structure-aware recurrent neural network agent is designed and seaml…
The goal of this paper is to make a strong point for the usage of dynamical models when using reinforcement learning (RL) for feedback control of dynamical systems governed by partial differential equations (PDEs). To breach the gap between the immense promises we see in RL and the applicability in complex engineering systems, the main challenges are the massive requirements in terms of the training data, as well as the lack of performance guarantees. We present a solution for the first issue using a data-driven surrogate model in the form of a convolutional LSTM with actuation. We demonstrate…
University of Alberta（阿尔伯塔大学）；Mitsubishi Electric Research Laboratories (MERL)（三菱电机研究实验室（MERL））；Vector Institute（向量研究所）
Recent work has shown that reinforcement learning (RL) is a promising approach to control dynamical systems described by partial differential equations (PDE). This paper shows how to use RL to tackle more general PDE control problems that have continuous high-dimensional action spaces with spatial relationship among action dimensions. In particular, we propose the concept of action descriptors, which encode regularities among spatially-extended action dimensions and enable the agent to control high-dimensional action PDEs. We provide theoretical evidence suggesting that this approach can be mo…
Tsinghua University（清华大学）；Louisiana State University（路易斯安那州立大学）
Learning to coordinate is a daunting problem in multi-agent reinforcement learning (MARL). Previous works have explored it from many facets, including cognition between agents, credit assignment, communication, expert demonstration, etc. However, less attention were paid to agents' decision structure and the hierarchy of coordination. In this paper, we explore the spatiotemporal structure of agents' decisions and consider the hierarchy of coordination from the perspective of multilevel emergence dynamics, based on which a novel approach, Learning to Advise and Learning from Advice (LALA), is p…
Reinforcement Learning (RL) has achieved impressive performance in many complex environments due to the integration with Deep Neural Networks (DNNs). At the same time, Genetic Algorithms (GAs), often seen as a competing approach to RL, had limited success in scaling up to the DNNs required to solve challenging tasks. Contrary to this dichotomic view, in the physical world, evolution and learning are complementary processes that continuously interact. The recently proposed Evolutionary Reinforcement Learning (ERL) framework has demonstrated mutual benefits to performance when combining the two …
Linear Partial Differential Equations (PDEs) govern the spatial-temporal dynamics of physical systems that are essential to building modern technology. When working with linear PDEs, designing a physical system for a specific outcome is difficult and costly due to slow and expensive explicit simulation of PDEs and the highly nonlinear relationship between a system's configuration and its behavior. In this work, we prove a parametric form that certain physical quantities in the Fourier domain must obey in linear PDEs, named the CZP (Constant-Zeros-Poles) framework. Applying CZP to antenna desig…
GE Research（通用电气研究院）；Binghamton University（宾汉姆顿大学）
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 trai…
SymPlex: A Structure-Aware Transformer for Symbolic PDE Solving
Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, USA（加州大学洛杉矶分校数学系）；Mechanical and Aerospace Engineering, University of California, Los Angeles, Los Angeles, CA, USA（加州大学洛杉矶分校机械与航空航天工程系）；Office of Advanced Research Computing (OARC), University of California, Los Angeles, Los Angeles, CA, USA（加州大学洛杉矶分校高级计算研究办公室）
Applications in quantitative finance such as optimal trade execution, risk management of options, and optimal asset allocation involve the solution of high dimensional and nonlinear Partial Differential Equations (PDEs). The connection between PDEs and systems of Forward-Backward Stochastic Differential Equations (FBSDEs) enables the use of advanced simulation techniques to be applied even in the high dimensional setting. Unfortunately, when the underlying application contains nonlinear terms, then classical methods both for simulation and numerical methods for PDEs suffer from the curse of di…
This paper addresses distributional offline continuous-time reinforcement learning (DOCTR-L) with stochastic policies for high-dimensional optimal control. A soft distributional version of the classical Hamilton-Jacobi-Bellman (HJB) equation is given by a semilinear partial differential equation (PDE). This `soft HJB equation' can be learned from offline data without assuming that the latter correspond to a previous optimal or near-optimal policy. A data-driven solution of the soft HJB equation uses methods of Neural PDEs and Physics-Informed Neural Networks developed in the field of Scientifi…
Neural networks (NNs) are often used as surrogates or emulators of partial differential equations (PDEs) that describe the dynamics of complex systems. A virtually negligible computational cost of such surrogates renders them an attractive tool for ensemble-based computation, which requires a large number of repeated PDE solves. Since the latter are also needed to generate sufficient data for NN training, the usefulness of NN-based surrogates hinges on the balance between the training cost and the computational gain stemming from their deployment. We rely on multi-fidelity simulations to reduc…
