Solving Bayesian Inverse Problems via Variational Autoencoders

Goh, Hwan; Sheriffdeen, Sheroze; Wittmer, Jonathan; Bui-Thanh, Tan

doi:10.48550/arxiv.1912.04212

Cited by 7 publications

(6 citation statements)

References 44 publications

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“…we seek a reduced order approximation in an optimal linear subspace [50,90,63,68,91,92]. If the problem does not allow such representation, nonlinear variants (e.g., autoencoders) could be considered as data compression tools [98,99,100,101]. Here, we prefer to employ POD because it is generally faster than the nonlinear variants.…”

Section: Proper Orthogonal Decomposition (Pod)mentioning

confidence: 99%

Data-driven reduced order modeling of poroelasticity of heterogeneous media based on a discontinuous Galerkin approximation

2021

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A simulation tool capable of speeding up the calculation for linear poroelasticity problems in heterogeneous porous media is of large practical interest for engineers, in particular, to effectively perform sensitivity analyses, uncertainty quantification, optimization, or control operations on the fluid pressure and bulk deformation fields. Towards this goal, we present here a non-intrusive model reduction framework using proper orthogonal decomposition (POD) and neural networks, based on the usual offline-online paradigm. As the conductivity of porous media can be highly heterogeneous and span several orders of magnitude, we utilize the interior penalty discontinuous Galerkin (DG) method as a full order solver to handle discontinuity and ensure local mass conservation during the offline stage. We then use POD as a data compression tool and compare the nested POD technique, in which time and uncertain parameter domains are compressed consecutively, to the classical POD method in which all domains are compressed simultaneously. The neural networks are finally trained to map the set of uncertain parameters, which could correspond to material properties, boundary conditions, or geometric characteristics, to the collection of coefficients calculated from an L 2 projection over the reduced basis. We then perform a non-intrusive evaluation of the neural networks to obtain coefficients corresponding to new values of the uncertain parameters during the online stage. We show that our framework provides reasonable approximations of the DG solution, but it is significantly faster. Moreover, the reduced order framework can capture sharp discontinuities of both displacement and pressure fields resulting from the heterogeneity in the media conductivity, which is generally challenging for intrusive reduced order methods. The sources of error are presented, showing that the nested POD technique is computationally advantageous and still provides comparable accuracy to the classical POD method. We also explore the effect of different choices of the hyperparameters of the neural network on the framework performance.Keywords poroelasticity • reduced order modeling • neural networks • discontinuous Galerkin • heterogeneity • finite element

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Section: Proper Orthogonal Decomposition (Pod)mentioning

confidence: 99%

Data-driven reduced order modeling of poroelasticity of heterogeneous media based on a discontinuous Galerkin approximation

2021

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“…Due to the benefit of being efficient, flexible and versatile, autoencoders are widely used in compressing the input data. It has shown some successful applications in the field of parameter estimation of PDEs [32,33,34,35,36,37,38]. Therefore, a convolutional AE is designed to compress the input power maps of the heat equations to be compact latent vectors.…”

Section: The Hybrid Framework Of Ae and Ig Based Networkmentioning

confidence: 99%

An unsupervised learning approach to solving heat equations on chip based on Auto Encoder and Image Gradient

He,

Pathak

2020

Preprint

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“…In addition to showing that a trained autoencoder has memory and computational advantages over existing methods, we develop an efficient data generation and training procedure based on the Bayesian inversion formulation that enables our approach to maintain its advantages, even when including the cost of training the autoencoder. While variations of autoencoders are often used for generative modeling (Kingma and Welling 2013), and others assign a physical interpretation to the latent space (Goh et al 2019), we use the autoencoding capabilities of the autoencoder architecture to allow the training process to find the most effective compressed latent representation for seismic input data. We provide a mathematical description of the FWI problem and the inverse problem in section 2.…”

Section: Introductionmentioning

confidence: 99%

An autoencoder compression approach for accelerating large-scale inverse problems

Wittmer,

Badger,

Sundar

et al. 2023

Inverse Problems

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PDE-constrained inverse problems are some of the most challenging and computationally demanding problems in computational science today. Fine meshes that are required to accurately compute the PDE solution introduce an enormous number of parameters and require large scale computing resources such as more processors and more memory to solve such systems in a reasonable time. For inverse problems constrained by time dependent PDEs, the adjoint method that is often employed to efﬁciently compute gradients and higher order derivatives requires solving a time-reversed, so-called adjoint PDE that depends on the forward PDE solution at each timestep. This necessitates the storage of a high dimensional forward solution vector at every timestep. Such a procedure quickly exhausts the available memory resources. Several approaches that trade additional computation for reduced memory footprint have been proposed to mitigate the memory bottleneck, including checkpointing and compression strategies. In this work, we propose a close-to-ideal scalable compression approach using autoencoders to eliminate the need for checkpointing and substantial memory storage, thereby reducing both the time-to-solution and memory requirements. We compare our approach with checkpointing and an off-the-shelf compression approach on an earth-scale ill-posed seismic inverse problem. The results verify the expected close-to-ideal speedup for both the gradient and Hessianvector product using the proposed autoencoder compression approach. To highlight the usefulness of the proposed approach, we combine the autoencoder compression with the data-informed active subspace (DIAS) prior to show how the DIAS method can be affordably extended to large scale problems without the need of checkpointing and large memory.

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Solving Bayesian Inverse Problems via Variational Autoencoders

Cited by 7 publications

References 44 publications

Data-driven reduced order modeling of poroelasticity of heterogeneous media based on a discontinuous Galerkin approximation

Data-driven reduced order modeling of poroelasticity of heterogeneous media based on a discontinuous Galerkin approximation

An unsupervised learning approach to solving heat equations on chip based on Auto Encoder and Image Gradient

An autoencoder compression approach for accelerating large-scale inverse problems

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