Bayesian deep convolutional encoder–decoder networks for surrogate modeling and uncertainty quantification

We are interested in the development of surrogate models for uncertainty quantification and propagation in problems governed by stochastic PDEs using a deep convolutional encoder–decoder network in a similar fashion to approaches considered in deep learning for image-to-image regression tasks. Since...

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Published in:	Journal of computational physics Vol. 366; pp. 415 - 447
Main Authors:	Zhu, Yinhao, Zabaras, Nicholas
Format:	Journal Article
Language:	English
Published:	Cambridge Elsevier Inc 01.08.2018 Elsevier Science Ltd
Subjects:	Algorithms Artificial neural networks Bayesian analysis Bayesian neural networks Computational physics Computer simulation Computer vision Convolutional encoder–decoder networks Deep learning Gaussian process Machine learning Neural networks Parameter uncertainty Permeability Porous media flows Propagation Regression models Statistical inference Stochastic models Training Uncertainty quantification Deep learning Uncertainty quantification Porous media flows Convolutional encoder–decoder networks Bayesian neural networks
ISSN:	0021-9991, 1090-2716
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Abstract	We are interested in the development of surrogate models for uncertainty quantification and propagation in problems governed by stochastic PDEs using a deep convolutional encoder–decoder network in a similar fashion to approaches considered in deep learning for image-to-image regression tasks. Since normal neural networks are data-intensive and cannot provide predictive uncertainty, we propose a Bayesian approach to convolutional neural nets. A recently introduced variational gradient descent algorithm based on Stein's method is scaled to deep convolutional networks to perform approximate Bayesian inference on millions of uncertain network parameters. This approach achieves state of the art performance in terms of predictive accuracy and uncertainty quantification in comparison to other approaches in Bayesian neural networks as well as techniques that include Gaussian processes and ensemble methods even when the training data size is relatively small. To evaluate the performance of this approach, we consider standard uncertainty quantification tasks for flow in heterogeneous media using limited training data consisting of permeability realizations and the corresponding velocity and pressure fields. The performance of the surrogate model developed is very good even though there is no underlying structure shared between the input (permeability) and output (flow/pressure) fields as is often the case in the image-to-image regression models used in computer vision problems. Studies are performed with an underlying stochastic input dimensionality up to 4225 where most other uncertainty quantification methods fail. Uncertainty propagation tasks are considered and the predictive output Bayesian statistics are compared to those obtained with Monte Carlo estimates. •Bayesian Convolutional Encoder–Decoder Deep Networks for Uncertainty Quantification Tasks.•Integrating Stein variational inference for exploring the high-dimensional posterior distribution of the network parameters.•Addressing the curse of dimensionality showing applications in porous media flows with permeability dimensionality of 4225.
AbstractList	We are interested in the development of surrogate models for uncertainty quantification and propagation in problems governed by stochastic PDEs using a deep convolutional encoder–decoder network in a similar fashion to approaches considered in deep learning for image-to-image regression tasks. Since normal neural networks are data-intensive and cannot provide predictive uncertainty, we propose a Bayesian approach to convolutional neural nets. A recently introduced variational gradient descent algorithm based on Stein's method is scaled to deep convolutional networks to perform approximate Bayesian inference on millions of uncertain network parameters. This approach achieves state of the art performance in terms of predictive accuracy and uncertainty quantification in comparison to other approaches in Bayesian neural networks as well as techniques that include Gaussian processes and ensemble methods even when the training data size is relatively small. To evaluate the performance of this approach, we consider standard uncertainty quantification tasks for flow in heterogeneous media using limited training data consisting of permeability realizations and the corresponding velocity and pressure fields. The performance of the surrogate model developed is very good even though there is no underlying structure shared between the input (permeability) and output (flow/pressure) fields as is often the case in the image-to-image regression models used in computer vision problems. Studies are performed with an underlying stochastic input dimensionality up to 4225 where most other uncertainty quantification methods fail. Uncertainty propagation tasks are considered and the predictive output Bayesian statistics are compared to those obtained with Monte Carlo estimates. We are interested in the development of surrogate models for uncertainty quantification and propagation in problems governed by stochastic PDEs using a deep convolutional encoder–decoder network in a similar fashion to approaches considered in deep learning for image-to-image regression tasks. Since normal neural networks are data-intensive and cannot provide predictive uncertainty, we propose a Bayesian approach to convolutional neural nets. A recently introduced variational gradient descent algorithm based on Stein's method is scaled to deep convolutional networks to perform approximate Bayesian inference on millions of uncertain network parameters. This approach achieves state of the art performance in terms of predictive accuracy and uncertainty quantification in comparison to other approaches in Bayesian neural networks as well as techniques that include Gaussian processes and ensemble methods even when the training data size is relatively small. To evaluate the performance of this approach, we consider standard uncertainty quantification tasks for flow in heterogeneous media using limited training data consisting of permeability realizations and the corresponding velocity and pressure fields. The performance of the surrogate model developed is very good even though there is no underlying structure shared between the input (permeability) and output (flow/pressure) fields as is often the case in the image-to-image regression models used in computer vision problems. Studies are performed with an underlying stochastic input dimensionality up to 4225 where most other uncertainty quantification methods fail. Uncertainty propagation tasks are considered and the predictive output Bayesian statistics are compared to those obtained with Monte Carlo estimates. •Bayesian Convolutional Encoder–Decoder Deep Networks for Uncertainty Quantification Tasks.•Integrating Stein variational inference for exploring the high-dimensional posterior distribution of the network parameters.•Addressing the curse of dimensionality showing applications in porous media flows with permeability dimensionality of 4225.
Author	Zhu, Yinhao Zabaras, Nicholas
Author_xml	– sequence: 1 givenname: Yinhao surname: Zhu fullname: Zhu, Yinhao email: yzhu10@nd.edu – sequence: 2 givenname: Nicholas surname: Zabaras fullname: Zabaras, Nicholas email: nzabaras@gmail.com
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Snippet	We are interested in the development of surrogate models for uncertainty quantification and propagation in problems governed by stochastic PDEs using a deep...
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SubjectTerms	Algorithms Artificial neural networks Bayesian analysis Bayesian neural networks Computational physics Computer simulation Computer vision Convolutional encoder–decoder networks Deep learning Gaussian process Machine learning Neural networks Parameter uncertainty Permeability Porous media flows Propagation Regression models Statistical inference Stochastic models Training Uncertainty quantification
Title	Bayesian deep convolutional encoder–decoder networks for surrogate modeling and uncertainty quantification
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