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1. A Neural Network Approach Applied to Multi-Agent Optimal Control

   Onken, D; Nurbekyan, L; Li, Xingjian; Wu Fung, S; Osher, S; Ruthotto, L (January 2021, European Control Conference)

   null (Ed.)

   We propose a neural network approach for solving high-dimensional optimal control problems. In particular, we focus on multi-agent control problems with obstacle and collision avoidance. These problems immediately become high-dimensional, even for moderate phase-space dimensions per agent. Our approach fuses the Pontryagin Maximum Principle and Hamilton-Jacobi-Bellman (HJB) approaches and parameterizes the value function with a neural network. Our approach yields controls in a feedback form for quick calculation and robustness to moderate disturbances to the system. We train our model using the objective function and optimality conditions of the control problem. Therefore, our training algorithm neither involves a data generation phase nor solutions from another algorithm. Our model uses empirically effective HJB penalizers for efficient training. By training on a distribution of initial states, we ensure the controls' optimality is achieved on a large portion of the state-space. Our approach is grid-free and scales efficiently to dimensions where grids become impractical or infeasible. We demonstrate our approach's effectiveness on a 150-dimensional multi-agent problem with obstacles. 

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2. A Neural Network Approach for Stochastic Optimal Control

   Li, Xingjian; Verma, Deepanshu; Ruthotto, Lars (June 2024, SIAM journal on scientific computing)

   We present a neural network approach for approximating the value function of high- dimensional stochastic control problems. Our training process simultaneously updates our value function estimate and identifies the part of the state space likely to be visited by optimal trajectories. Our approach leverages insights from optimal control theory and the fundamental relation between semi-linear parabolic partial differential equations and forward-backward stochastic differential equations. To focus the sampling on relevant states during neural network training, we use the stochastic Pontryagin maximum principle (PMP) to obtain the optimal controls for the current value function estimate. By design, our approach coincides with the method of characteristics for the non-viscous Hamilton-Jacobi-Bellman equation arising in deterministic control problems. Our training loss consists of a weighted sum of the objective functional of the control problem and penalty terms that enforce the HJB equations along the sampled trajectories. Importantly, training is unsupervised in that it does not require solutions of the control problem. Our numerical experiments highlight our scheme’s ability to identify the relevant parts of the state space and produce meaningful value estimates. Using a two-dimensional model problem, we demonstrate the importance of the stochastic PMP to inform the sampling and compare to a finite element approach. With a nonlinear control affine quadcopter example, we illustrate that our approach can handle complicated dynamics. For a 100-dimensional benchmark problem, we demonstrate that our approach improves accuracy and time-to-solution and, via a modification, we show the wider applicability of our scheme. 

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3. Scaling Learning-based Policy Optimization for Temporal Logic Tasks by Controller Network Dropout

   https://doi.org/10.1145/3696112  

   Hashemi, Navid; Hoxha, Bardh; Prokhorov, Danil; Fainekos, Georgios; Deshmukh, Jyotirmoy V (October 2024, ACM Transactions on Cyber-Physical Systems)

   This article introduces a model-based approach for training feedback controllers for an autonomous agent operating in a highly non-linear (albeit deterministic) environment. We desire the trained policy to ensure that the agent satisfies specific task objectives and safety constraints, both expressed in Discrete-Time Signal Temporal Logic (DT-STL). One advantage for reformulation of a task via formal frameworks, like DT-STL, is that it permits quantitative satisfaction semantics. In other words, given a trajectory and a DT-STL formula, we can compute therobustness, which can be interpreted as an approximate signed distance between the trajectory and the set of trajectories satisfying the formula. We utilize feedback control, and we assume a feed forward neural network for learning the feedback controller. We show how this learning problem is similar to training recurrent neural networks (RNNs), where the number of recurrent units is proportional to the temporal horizon of the agent’s task objectives. This poses a challenge: RNNs are susceptible to vanishing and exploding gradients, and naïve gradient descent-based strategies to solve long-horizon task objectives thus suffer from the same problems. To address this challenge, we introduce a novel gradient approximation algorithm based on the idea of dropout or gradient sampling. One of the main contributions is the notion ofcontroller network dropout, where we approximate the NN controller in several timesteps in the task horizon by the control input obtained using the controller in a previous training step. We show that our control synthesis methodology can be quite helpful for stochastic gradient descent to converge with less numerical issues, enabling scalable back-propagation over longer time horizons and trajectories over higher-dimensional state spaces. We demonstrate the efficacy of our approach on various motion planning applications requiring complex spatio-temporal and sequential tasks ranging over thousands of timesteps. 

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4. An offline multi‐scale unsaturated poromechanics model enabled by self‐designed/self‐improved neural networks

   https://doi.org/10.1002/nag.3196  

   Heider, Yousef; Suh, Hyoung_Suk; Sun, WaiChing (February 2021, International Journal for Numerical and Analytical Methods in Geomechanics)

   Abstract Supervised machine learning via artificial neural network (ANN) has gained significant popularity for many geomechanics applications that involves multi‐phase flow and poromechanics. For unsaturated poromechanics problems, the multi‐physics nature and the complexity of the hydraulic laws make it difficult to design the optimal setup, architecture, and hyper‐parameters of the deep neural networks. This paper presents a meta‐modeling approach that utilizes deep reinforcement learning (DRL) to automatically discover optimal neural network settings that maximize a pre‐defined performance metric for the machine learning constitutive laws. This meta‐modeling framework is cast as a Markov Decision Process (MDP) with well‐defined states (subsets of states representing the proposed neural network (NN) settings), actions, and rewards. Following the selection rules, the artificial intelligence (AI) agent, represented in DRL via NN, self‐learns from taking a sequence of actions and receiving feedback signals (rewards) within the selection environment. By utilizing the Monte Carlo Tree Search (MCTS) to update the policy/value networks, the AI agent replaces the human modeler to handle the otherwise time‐consuming trial‐and‐error process that leads to the optimized choices of setup from a high‐dimensional parametric space. This approach is applied to generate two key constitutive laws for the unsaturated poromechanics problems: (1) the path‐dependent retention curve with distinctive wetting and drying paths. (2) The flow in the micropores, governed by an anisotropic permeability tensor. Numerical experiments have shown that the resultant ML‐generated material models can be integrated into a finite element (FE) solver to solve initial‐boundary‐value problems as replacements of the hand‐craft constitutive laws. 

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5. Learning a Decentralized Multi-arm Motion Planner

   Ha, Huy; Xu, Jingxi; Song, Shuran (November 2020, Proceedings of the 2020 Conference on Robot Learning)

   We present a closed-loop multi-arm motion planner that is scalable and flexible with team size. Traditional multi-arm robotic systems have relied on centralized motion planners, whose run times often scale exponentially with team size, and thus, fail to handle dynamic environments with open-loop control. In this paper, we tackle this problem with multi-agent reinforcement learning, where a shared policy network is trained to control each individual robot arm to reach its target end-effector pose given observations of its workspace state and target end-effector pose. The policy is trained using Soft Actor-Critic with expert demonstrations from a sampling-based motion planning algorithm (i.e., BiRRT). By leveraging classical planning algorithms, we can improve the learning efficiency of the reinforcement learning algorithm while retaining the fast inference time of neural networks. The resulting policy scales sub-linearly and can be deployed on multi-arm systems with variable team sizes. Thanks to the closed-loop and decentralized formulation, our approach generalizes to 5-10 multiarm systems and dynamic moving targets (>90% success rate for a 10-arm system), despite being trained on only 1-4 arm planning tasks with static targets. 

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