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Managing engineering systems with large state and action spaces through deep reinforcement learning

C. P. Andriotis, K. G. Papakonstantinou

arXiv:1811.02052v1eess.SYcs.LGcs.MA

TL;DR

Large multi-component engineering systems create exponentially large state and action spaces that make conventional model-based decision procedures difficult to apply. The paper introduces DCMAC, an off-policy actor-critic DRL framework with factorized component and subsystem actions and a centralized value function. Across the reported evaluations, DCMAC finds complex policies and outperforms optimized engineering baselines.

  • Problem

    Large multi-component systems have state and action spaces that scale exponentially with component count, while explicit whole-system dynamics may be difficult to obtain.

  • Method

    DCMAC is an off-policy actor-critic DRL approach that factorizes actions into component- and subsystem-level decisions while maintaining a centralized system value function.

  • Results

    DCMAC outperforms optimized baselines based on time-based, condition-based, and periodic policies across the reported engineering-system settings.

  • Takeaways & Limitations

    DCMAC provides efficient life-cycle policies and individualized decisions for challenging high-dimensional multi-component engineering systems.

  • Takeaways & Limitations

    State and observation handling assumes known component states or observations up to known precision, and alternative reductions may rely on component similarity or engineered macro-states.

Abstract

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Decision-making for engineering systems can be efficiently formulated as a Markov Decision Process (MDP) or a Partially Observable MDP (POMDP). Typical MDP and POMDP solution procedures utilize offline knowledge about the environment and provide detailed policies for relatively small systems with tractable state and action spaces. However, in large multi-component systems the sizes of these spaces easily explode, as system states and actions scale exponentially with the number of components, whereas environment dynamics are difficult to be described in explicit forms for the entire system and may only be accessible through numerical simulators. In this work, to address these issues, an integrated Deep Reinforcement Learning (DRL) framework is introduced. The Deep Centralized Multi-agent Actor Critic (DCMAC) is developed, an off-policy actor-critic DRL approach, providing efficient life-cycle policies for large multi-component systems operating in high-dimensional spaces. Apart from deep function approximations that parametrize large state spaces, DCMAC also adopts a factorized representation of the system actions, being able to designate individualized component- and subsystem-level decisions, while maintaining a centralized value function for the entire system. DCMAC compares well against Deep Q-Network (DQN) solutions and exact policies, where applicable, and outperforms optimized baselines that are based on time-based, condition-based and periodic policies.

1. Introduction

Large engineering systems make conventional maintenance and inspection planning difficult because component interactions create enormous state and action spaces. The paper develops DRL methods, especially DCMAC, to provide scalable life-cycle policies without relying on simplified system representations.

  • Motivation: Multi-component interactions can make component-level optimality insufficient for system-level maintenance and inspection decisions.Different component-state combinations may require coordinated actions across system parts.
  • Motivation: Conventional MDP and POMDP methods become impractical as large systems make state and action spaces scale exponentially with component count.Transition matrices and action-value evaluations also become extremely large or computationally expensive.
  • Existing approaches: Modeling simplifications such as exploiting component similarity or engineering macro-states can reduce complexity, but may trade accuracy or solution quality.Similarity assumptions are most suitable for homogeneous, structurally independent components.
  • Contribution: The paper investigates DQN and deep policy-gradient architectures and introduces DCMAC for stochastic control of large engineering systems.DCMAC uses deep actor and critic networks trained with environment signals and replayed transitions.
  • Contribution: DCMAC provides individualized component- and subsystem-level decisions while maintaining a centralized value function for the entire system.The approach is designed for high-dimensional state and action spaces in otherwise practically intractable multi-component problems.
  • Evaluation: The study evaluates DRL methods against exact policies where applicable and optimized baseline policies.The baselines include standard engineering-policy comparisons described in the paper.

2. Markov Decision Processes

MDPs model sequential decisions using states, actions, transitions, rewards, and policies, while POMDPs replace exact state information with beliefs. These classical formulations can require explicit environment models, motivating reinforcement-learning alternatives that learn from interaction.

  • Markov Decision Processes: An MDP consists of states, stochastic transitions, rewards, and actions, with the agent selecting actions and receiving rewards as the environment evolves.The current state and action determine the transition distribution under the Markov assumption.
  • Markov Decision Processes: The Markov property assumes the current state and selected action are sufficient statistics for the next state, regardless of prior history.Non-Markovian environments can be transformed through state augmentation techniques.
  • Policies: Policies map states to actions deterministically or to action probabilities stochastically.For discrete actions, a stochastic policy is represented by probabilities over available actions.
  • Objective functions: The discounted return aggregates rewards from the current decision step through the planning horizon, with γ controlling the relative weight of future rewards.γ=0 weights only the current decision, whereas γ=1 weights decisions equally across the horizon.
  • Value functions: Action-value and value functions represent expected returns conditioned on a state-action pair or a state, respectively.The action-value function has a recursive Bellman form involving immediate reward and future value.
  • Partially Observable MDPs: POMDPs handle partial observability by replacing exact states with beliefs updated from actions, observations, and conditional probability models.Beliefs act as alternative states, but their continuous simplex makes POMDPs harder to solve.
  • Reinforcement learning: Reinforcement learning removes the need for offline explicit environment models by learning from direct interaction and temporal-difference samples.RL algorithms may be on-policy or off-policy, and policy-gradient methods directly update parameterized policies.

3. Deep Reinforcement Learning

Deep reinforcement learning replaces cumbersome tabular value and policy representations with neural networks. The paper reviews DQN and policy-gradient methods, including replay, target networks, actor-critic estimation, and off-policy corrections.

  • Deep function approximation: Neural-network function approximation reduces high-dimensional value or policy evaluation to determining parameters rather than tabulating every state.The approximated functions include Q-functions, value functions, and policies.
  • Deep Q-Networks: DQN parametrizes the action-value function with a deep network that maps the state to Q-values for available actions.Its training objective minimizes a loss between predicted Q-values and target values.
  • Deep Q-Networks: Experience replay trains DQN from randomly sampled past transition tuples, while a separate target network supports training stability.The target network is updated more slowly than the original network.
  • Deep Q-Networks: Double Q-learning uses the original network to select actions and the target network to evaluate them, reducing overoptimistic value estimates.The paper describes this modification as improving training stability and near-optimal solutions.
  • Deep policy gradients: Deep policy-gradient methods parameterize a policy network and update it using gradients weighted by action-value or advantage estimates.Actor-critic methods train the policy approximator with a value approximator.
  • Actor-critic methods: Actor-critic architectures use an actor for policy approximation and a critic for estimating a value or action-value function.An action-value critic receives both the state and selected action as inputs.
  • Off-policy learning: Off-policy policy-gradient estimators can reuse samples from a behavior policy through importance sampling, but importance weights may have high variance.Truncated importance sampling is presented as a way to manage this variance.

4. Deep Centralized Multi-agent Actor Critic

DCMAC addresses large engineering-system control problems by combining deep reinforcement learning with factorized component- and subsystem-level actions and a centralized critic. Its architecture reduces the actor’s output dimensionality while preserving system-level value evaluation and supports off-policy learning for simulation-expensive environments.

  • Action representation: DCMAC factorizes system actions across individualized component and subsystem control units, assuming conditional independence given the full system state.The action representation is designed to alleviate output-layer complexity in large action spaces.
  • Action representation: A system with 540 possible joint actions is represented by an 18-dimensional DCMAC actor output without loss of generality.The example has five control units with 3, 4, 3, 5, and 3 available actions, respectively.
  • Centralized architecture: The actor uses shared parameters and receives the entire system state, allowing each control unit to account for other agents’ states and actions.The agents are supported by a centralized actor rather than independent networks.
  • Centralized architecture: The centralized critic approximates the value function over the entire system space and supplies a global measure for policy updates.Critic parameters are updated through mean squared error.
  • Training: DCMAC uses weighted-advantage backpropagation with importance sampling to update its off-policy actor from replayed experience.The approach can improve sample efficiency when experience comes from computationally expensive numerical simulations, although products of importance weights may increase estimator variance.
  • Partial observability: For partially observed engineering systems, the same DRL equations can operate on continuously updated beliefs instead of directly observed states when transition and observation models are known.This belief-MDP formulation uses component transition and observation models and retains the DRL equations with states replaced by beliefs.

5. Numerical experiments

Numerical experiments evaluate DCMAC across stationary and non-stationary engineering systems, including fully and partially observable settings and a simulated truss bridge. DCMAC generally converges effectively, matches or approaches exact policies, and outperforms optimized baseline policies in the reported tests.

  • Experimental scope: The experiments cover a stationary parallel-series MDP, a non-stationary k-out-of-n system under MDP and POMDP settings, and a corrosion-deteriorating truss bridge POMDP.The systems vary in state-space size, component dependencies, transition stationarity, available actions, observation accuracy, and inspection choices.
  • System I: System I contains 5 deteriorating components with 4 damage states and two component actions: do nothing or replace.System failure depends on pertinent combinations of failed components, and component state costs receive a factor-of-24 penalty after failure.
  • System I: DCMAC and DQN both converge to the System I exact solution with less than 5% error, while DCMAC has better anytime performance and faster convergence.Across policy realizations, action agreement with the exact policy is about 99% for DCMAC and 96% for DQN.
  • System II: System II uses 10 components with 4 states each, non-stationary transitions, four actions per component, and costs evaluated under varying observability.The actions are do nothing, minor repair, major repair, and replacement; major repair also reduces deterioration rate, while replacement restores the initial state and rate.
  • System II: Across System II observability levels, DCMAC discovers substantially better policies than the optimized baselines, including cases where baselines have better observability.The advantage is achieved before 3.0 thousand episodes except for TCBM-II under perfect observability, which is surpassed after 4.4 thousand episodes.
  • System III: For the truss bridge, DCMAC reduces life-cycle cost by 5.8% to 22.3% relative to optimized fixed-period inspection baselines.The baselines use periodic inspections every 2, 5, 10, or 15 years and component-uniform thresholds; DCMAC also favors major repairs because they reduce deterioration rate.

6. Conclusions

The paper introduces DCMAC for high-dimensional engineering-system control, combining centralized value estimation with factorized component actions. Across challenging domains, it is reported to find competent near-optimal policies and outperform optimized practice-based baselines.

  • 6. Conclusions: DCMAC is an off-policy DRL approach for large multi-component engineering systems with immense state and action spaces.It is presented as the paper’s first DRL implementation for this control setting.
  • 6. Conclusions: A centralized value function and actor network use factorized action probabilities to represent component- and subsystem-level decisions.The architecture establishes conditional independences so actor output dimensions scale linearly with the number of control units.
  • 6. Conclusions: DCMAC handles high-dimensional, non-stationary, multi-state, multi-component environments under partial or full observability.The reported applications include generic deteriorating systems and structural systems.
  • 6. Conclusions: DCMAC finds intelligent solutions and complex decision policies while outperforming optimized baselines reflecting standard engineering practice.The paper reports this pattern across various problem settings.

Appendix A. DQN Algorithm

The appendix describes a Double Deep Q-Network implementation for fully observable MDP environments. The algorithm selects actions using an online Q-network, learns from replayed experience, and periodically updates a target network.

  • Appendix A. DQN Algorithm: DDQN is used in this paper only for System I, where complete observability is assumed.The appendix notes that partial observability and belief MDPs can be applied through the main algorithm.
  • Appendix A. DQN Algorithm: At each step, the algorithm selects an exploratory random action or the action maximizing Q(s_t,a) under the online network.The action-selection rule is shown in Algorithm A1.
  • Appendix A. DQN Algorithm: The agent collects rewards and successor states, then stores experience tuples in a replay buffer.Batches of stored tuples are sampled for subsequent updates.
  • Appendix A. DQN Algorithm: Target values use the immediate reward for terminal transitions and a discounted next-state estimate otherwise.The update distinguishes terminal and nonterminal sampled transitions.
  • Appendix A. DQN Algorithm: The Q-network parameters are updated by gradient descent, while target-network weights are periodically synchronized with the Q-network.Algorithm A1 applies the target update when the prescribed update time is reached.

B.1. Systems I and II

Systems I and II share core component specifications, but System II introduces deterioration-rate-dependent, non-stationary transitions and partially observable component states. Its action transition matrices incorporate maintenance effects and probabilistic action success.

  • B.1. Systems I and II: The five components of System I and the first five components of System II share damage states and maintenance costs, while System II has non-stationary transitions.The systems otherwise differ through transition dynamics that depend on deterioration rate.
  • B.1. Systems I and II: System II transition matrices are obtained for each deterioration rate by interpolating initial and final transition matrices.The transition probabilities are shown as functions of deterioration rate, whereas System I uses stationary probabilities.
  • B.1. Systems I and II: The transition matrices and observation model jointly represent deterioration-rate-dependent dynamics and imperfect component-state information.The appendix defines these elements for the System II environment.
  • B.1. Systems I and II: Actions 2 and 3 have identical damage-state effects, but action 3 additionally reduces deterioration rate by 5 steps.Both actions have a success rate of 0.95; with probability 0.05, the environment follows the uncontrolled transition.
  • B.1. Systems I and II: The observation model uses a constant precision p across actions, assigning probability p to the correct component state while allowing adjacent-state observations.Under perfect observation, the observation matrix becomes the identity matrix and the true state receives probability 1.

B.2. System III

System III models steel-truss deterioration through stochastic corrosion-driven section loss, discretized into component damage states. The resulting transition and cost models support a large structural maintenance problem with non-stationary damage evolution.

  • B.2. System III: Steel corrosion is modeled through mean uniform section loss, with penetration represented as a power law of exposure time.The section-loss percentage is treated stochastically using a gamma process.
  • B.2. System III: Gamma-process increments describe stochastic deterioration between time steps and can represent continuous Markovian transitions in discrete time.The increment between two times follows a gamma distribution.
  • B.2. System III: With β=1.0 the gamma process is stationary, whereas the example uses β=1.5 to represent non-stationary damage evolution in highly corrosive environments.The parameters are calibrated using a mean section loss of 40% and standard deviation of 7.5% in 70 years.
  • B.2. System III: The continuous section-loss space is discretized in 2.5% steps, and members exceeding 60% loss are considered failed.This produces 25 damage states per component and 25x25 transition matrices for each deterioration rate.
  • B.2. System III: Component deterioration causes direct losses proportional to steel volume loss, while maintenance costs are proportional to initial member volume.The appendix specifies maximum values for failure losses and maintenance actions.

Appendix C. Deep networks

The appendix describes the deep-network inputs, architectures, training procedures, and hyperparameter considerations used in the examples. It also notes that architecture and hyperparameter optimization is outside the work’s scope.

  • Inputs: Network inputs combine the finite-horizon time index with all component states, including damage states and deterioration rates for non-stationary problems.Component damage states use one-zero vectorized representations, while partial observability uses probability-distribution vectors.
  • Network architectures: DCMAC uses the same 40x40 fully connected hidden-layer architecture for its actor and critic networks, with ReLU hidden-layer activations.In System I, DQN uses two fully connected layers of 40 dimensions each, and its output approximates 32 action-related Q-functions.
  • Training: Training uses batch-based sample estimates of gradients and first-order stochastic optimization methods.The appendix describes batch training as central to deep learning because it facilitates faster training while retaining convergence properties.
  • Training: Actor learning rates range from 10^-4 to 10^-5 and critic learning rates from 10^-3 to 10^-4, with batch size 32 and replay buffers of 2-3·10^5 samples.Learning rates begin higher and are adjusted downward during training; DCMAC truncates importance-sampling weights at c = 2 to reduce gradient-estimator variance.
  • Scope: Hyperparameters can affect convergence time and, in some cases, the quality of the converged solution, but optimized network specifications are beyond this work’s scope.The final example specifications resulted from experimentation, while comparable solutions may be obtained through finer tuning.
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