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

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Harbir Antil , Howard C. Elman , Akwum Onwunta , Deepanshu Verma ·

YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.1088/2632-2153/ace67c

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.

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Partial Differential Equation

Computer Science

Algorithm

Mathematics

Artificial Intelligence

Machine Learning

Mathematical Analysis

Quantum Mechanics

Physics

Geometry

Concepts

Markov chain Monte Carlo Parameterized complexity Nonlinear system Inverse problem Partial differential equation Computer science Applied mathematics Artificial neural network Mathematical optimization Bayesian probability Markov chain Variable-order Bayesian network Algorithm Inverse Mathematics Artificial intelligence Bayesian inference Machine learning Mathematical analysis Quantum mechanics Physics Geometry

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Type: article
Language: en
Landing Page: https://doi.org/10.1088/2632-2153/ace67c
PDF: https://iopscience.iop.org/article/10.1088/2632-2153/ace67c/pdf
OA Status: gold
Cited By: 4
References: 66
Related Works: 10
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DOI: https://doi.org/10.1088/2632-2153/ace67c

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

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Language: en

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Publication year: 2023

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Publication date: 2023-07-11

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Authors: Harbir Antil, Howard C. Elman, Akwum Onwunta, Deepanshu Verma

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Landing page: https://doi.org/10.1088/2632-2153/ace67c

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Concepts: Markov chain Monte Carlo, Parameterized complexity, Nonlinear system, Inverse problem, Partial differential equation, Computer science, Applied mathematics, Artificial neural network, Mathematical optimization, Bayesian probability, Markov chain, Variable-order Bayesian network, Algorithm, Inverse, Mathematics, Artificial intelligence, Bayesian inference, Machine learning, Mathematical analysis, Quantum mechanics, Physics, Geometry

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Cited by: 4

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