A deep neural network approach for parameterized PDEs and Bayesian inverse problems

Abstract We consider the simulation of Bayesian statistical inverse problems governed by large-scale linear and nonlinear partial differential equations (PDEs). Markov chain Monte Carlo (MCMC) algorithms are standard techniques to solve such problems. However, MCMC techniques are computationally cha...

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Bibliographic Details
Published in:Machine learning: science and technology 2023-09, Vol.4 (3), p.35015
Main Authors: Antil, Harbir, Elman, Howard C, Onwunta, Akwum, Verma, Deepanshu
Format: Article
Language:English
Subjects:
Algorithms
Artificial neural networks
Bayesian analysis
deep neural network
Fluid flow
fractional time derivative
Inverse problems
Markov chains
Mathematical analysis
metropolis-hastings
Navier-Stokes equations
Navier–Stokes
Nonlinear differential equations
Partial differential equations
statistical inverse problem
uncertainty quantification
Vortex shedding
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Summary:Abstract We consider the simulation of Bayesian statistical inverse problems governed by large-scale linear and nonlinear partial differential equations (PDEs). Markov chain Monte Carlo (MCMC) algorithms are standard techniques to solve such problems. However, MCMC techniques are computationally challenging as they require a prohibitive number of forward PDE solves. The goal of this paper is to introduce a fractional deep neural network (fDNN) based approach for the forward solves within an MCMC routine. Moreover, we discuss some approximation error estimates. We illustrate the efficiency of fDNN on inverse problems governed by nonlinear elliptic PDEs and the unsteady Navier–Stokes equations. In the former case, two examples are discussed, respectively depending on two and 100 parameters, with significant observed savings. The unsteady Navier–Stokes example illustrates that fDNN can outperform existing DNNs, doing a better job of capturing essential features such as vortex shedding.
ISSN:2632-2153
2632-2153
DOI:10.1088/2632-2153/ace67c