Multi-agent Reinforcement Learning Accelerated MCMC on Multiscale Inversion Problem

Chung, Eric; Efendiev, Yalchin; Leung, Wing Tat; Pun, Sai-Mang; Zhang, Zecheng

doi:10.48550/arxiv.2011.08954

Cited by 10 publications

(17 citation statements)

References 35 publications

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“…The errors are shown in Figure (15), we can see that the solution is very stable and accurate. If we look at Table (2), the ũH,2 indeed corrects u H,1 and improves the solution.…”

Section: Coarse Solution Assimilation For the Nonlinear Examplementioning

confidence: 85%

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HEI: hybrid explicit-implicit learning for multiscale problems

Efendiev,

Leung,

Lin

et al. 2021

Preprint

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Splitting method is a powerful method to handle application problems by splitting physics, scales, domain, and so on. Many splitting algorithms have been designed for efficient temporal discretization. In this paper, our goal is to use temporal splitting concepts in designing machine learning algorithms and, at the same time, help splitting algorithms by incorporating data and speeding them up. Since the spitting solution usually has an explicit and implicit part, we will call our method hybrid explicit-implict (HEI) learning. We will consider a recently introduced multiscale splitting algorithms, where the multiscale problem is solved on a coarse grid. To approximate the dynamics, only a few degrees of freedom are solved implicitly, while others explicitly. This splitting concept allows identifying degrees of freedom that need implicit treatment. In this paper, we use this splitting concept in machine learning and propose several strategies. First, the implicit part of the solution can be learned as it is more difficult to solve, while the explicit part can be computed. This provides a speed-up and data incorporation for splitting approaches. Secondly, one can design a hybrid neural network architecture because handling explicit parts requires much fewer communications among neurons and can be done efficiently. Thirdly, one can solve the coarse grid component via PDEs or other approximation methods and construct simpler neural networks for the explicit part of the solutions. We discuss these options and implement one of them by interpreting it as a machine translation task. This interpretation of the splitting scheme successfully enables us using the Transformer since it can perform model reduction for multiple time series and learn the connection between them. We also find that the splitting scheme is a great platform to predict the coarse solution with insufficient information of the target model: the target problem is partially given and we need to solve it through a known problem which approximates the target. Our machine learning model can incorporate and encode the given information from two different problems and then solve the target problems. We conduct four numerical examples and the results show that our method is stable and accurate.

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“…The errors are shown in Figure (15), we can see that the solution is very stable and accurate. If we look at Table (2), the ũH,2 indeed corrects u H,1 and improves the solution.…”

Section: Coarse Solution Assimilation For the Nonlinear Examplementioning

confidence: 85%

“…One of the contributions of this paper is the use of machine learning to accelerate the computations and discuss a possible design for machine learning [4,28,2,22,21]. We again comment that the computation of u 1 is more computationally difficult compared to u 2 as it uses implicit discretization.…”

Section: Introductionmentioning

confidence: 97%

HEI: hybrid explicit-implicit learning for multiscale problems

Efendiev,

Leung,

Lin

et al. 2021

Preprint

Self Cite

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show abstract

“…One choice is to estimate each state p k (x) given a sequence of observations up to k in time by the posterior distribution. This task of sequential prediction based on the online observations can then be categorized as the standard Bayesian filtering problems which have been thoroughly studied in the control theory [10,5,24,20]. This motivates us to formulate the inversion problem in the Bayesian framework.…”

Section: Bayesian Formulation and Outlinementioning

confidence: 99%

Theoretical and numerical studies of inverse source problem for the linear parabolic equation with sparse boundary measurements

Lin

Zhang²,

Zhang³

2021

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We consider the inverse source problem in the parabolic equation, where the unknown source possesses the semi-discrete formulation. Theoretically, we prove that the flux data from any nonempty open subset of the boundary can uniquely determine the semi-discrete source. This means the observed area can be extremely small, and that is why we call the data as sparse boundary data. For the numerical reconstruction, we formulate the problem from the Bayesian sequential prediction perspective and conduct the numerical examples which estimate the space-time-dependent source state by state. To better demonstrate the performance of the method, we solve two common multiscale problems from two models with a long sequence of the source. The numerical results illustrate that the inversion is accurate and efficient.

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“…For example, in petroleum engineering applications, engineers seek to calculate, using the porous media equations, the pressure field of the oil based on its permeability. In practice, the value of the oil permeability varies frequently; hence, one needs to calculate the oil pressure field for a distribution of fast-varying permeabilities [29,9,11]. The traditional numerical frameworks require intense computations to solve these parametric PDEs.…”

Section: Introductionmentioning

confidence: 99%

Accelerated replica exchange stochastic gradient Langevin diffusion enhanced Bayesian DeepONet for solving noisy parametric PDEs

Lin¹,

Moya²,

Zhang³

2021

Preprint

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The Deep Operator Networks (DeepONet) is a fundamentally different class of neural networks that we train to approximate nonlinear operators, including the solution operator of parametric partial differential equations (PDE). DeepONets have shown remarkable approximation and generalization capabilities even when trained with relatively small datasets. However, the performance of DeepONets deteriorates when the training data is polluted with noise, a scenario that occurs very often in practice. To enable DeepONets training with noisy data, we propose using the Bayesian framework of replica-exchange Langevin diffusion. Such a framework uses two particles, one for exploring and another for exploiting the loss function landscape of DeepONets. We show that the proposed framework's exploration and exploitation capabilities enable (1) improved training convergence for DeepONets in noisy scenarios and (2) attaching an uncertainty estimate for the predicted solutions of parametric PDEs. In addition, we show that replica-exchange Langeving Diffusion (remarkably) also improves the DeepONet's mean prediction accuracy in noisy scenarios compared with vanilla Deep-ONets trained with state-of-the-art gradient-based optimization algorithms (e.g., Adam). To reduce the potentially high computational cost of replica, in this work, we propose an accelerated training framework for replica-exchange Langevin diffusion that exploits the neural network architecture of DeepONets to reduce its computational cost up to 25% without compromising the proposed framework's performance. Finally, we illustrate the effectiveness of the proposed Bayesian framework using a series of experiments on four parametric PDE problems.

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Multi-agent Reinforcement Learning Accelerated MCMC on Multiscale Inversion Problem

Cited by 10 publications

References 35 publications

HEI: hybrid explicit-implicit learning for multiscale problems

HEI: hybrid explicit-implicit learning for multiscale problems

Theoretical and numerical studies of inverse source problem for the linear parabolic equation with sparse boundary measurements

Accelerated replica exchange stochastic gradient Langevin diffusion enhanced Bayesian DeepONet for solving noisy parametric PDEs

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