Structured Bayesian Gaussian process latent variable model: applications to data-driven dimensionality reduction and high-dimensional inversion

We introduce a methodology for nonlinear inverse problems using a variational Bayesian approach where the unknown quantity is a spatial field. A structured Bayesian Gaussian process latent variable model is used both to construct a low-dimensional generative model of the sample-based stochastic prior as well as a surrogate for the forward evaluation. Its Bayesian formulation captures epistemic uncertainty introduced by the limited number of input and output examples, automatically selects an appropriate dimensionality for the learned latent representation of the data, and rigorously propagates the uncertainty of the data-driven dimensionality reduction of the stochastic space through the forward model surrogate. The structured Gaussian process model explicitly leverages spatial information for an informative generative prior to improve sample efficiency while achieving computational tractability through Kronecker product decompositions of the relevant kernel matrices. Importantly, the Bayesian inversion is carried out by solving a variational optimization problem, replacing traditional computationally-expensive Monte Carlo sampling. The methodology is demonstrated on an elliptic PDE and is shown to return well-calibrated posteriors and is tractable with latent spaces with over 100 dimensions

Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks

In recent years, deep learning has proven to be a viable methodology for surrogate modeling and uncertainty quantification for a vast number of physical systems. However, in their traditional form, such models can require a large amount of training data. This is of particular importance for various engineering and scientific applications where data may be extremely expensive to obtain. To overcome this shortcoming, physics-constrained deep learning provides a promising methodology as it only utilizes the governing equations. In this work, we propose a novel auto-regressive dense encoder-decoder convolutional neural network to solve and model non-linear dynamical systems without training data at a computational cost that is potentially magnitudes lower than standard numerical solvers. This model includes a Bayesian framework that allows for uncertainty quantification of the predicted quantities of interest at each time-step. We rigorously test this model on several non-linear transient partial differential equation systems including the turbulence of the Kuramoto-Sivashinsky equation, multi-shock formation and interaction with 1D Burgers' equation and 2D wave dynamics with coupled Burgers' equations. For each system, the predictive results and uncertainty are presented and discussed together with comparisons to the results obtained from traditional numerical analysis methods.Comment: 48 pages, 30 figures, Accepted to Journal of Computational Physic

Quantifying model form uncertainty in Reynolds-averaged turbulence models with Bayesian deep neural networks

Data-driven methods for improving turbulence modeling in Reynolds-Averaged Navier-Stokes (RANS) simulations have gained significant interest in the computational fluid dynamics community. Modern machine learning algorithms have opened up a new area of black-box turbulence models allowing for the tuning of RANS simulations to increase their predictive accuracy. While several data-driven turbulence models have been reported, the quantification of the uncertainties introduced has mostly been neglected. Uncertainty quantification for such data-driven models is essential since their predictive capability rapidly declines as they are tested for flow physics that deviate from that in the training data. In this work, we propose a novel data-driven framework that not only improves RANS predictions but also provides probabilistic bounds for fluid quantities such as velocity and pressure. The uncertainties capture both model form uncertainty as well as epistemic uncertainty induced by the limited training data. An invariant Bayesian deep neural network is used to predict the anisotropic tensor component of the Reynolds stress. This model is trained using Stein variational gradient decent algorithm. The computed uncertainty on the Reynolds stress is propagated to the quantities of interest by vanilla Monte Carlo simulation. Results are presented for two test cases that differ geometrically from the training flows at several different Reynolds numbers. The prediction enhancement of the data-driven model is discussed as well as the associated probabilistic bounds for flow properties of interest. Ultimately this framework allows for a quantitative measurement of model confidence and uncertainty quantification for flows in which no high-fidelity observations or prior knowledge is available.Comment: 47 pages, 21 figures, Accepted to the Journal of Computational Physic

Predictive Collective Variable Discovery with Deep Bayesian Models

Extending spatio-temporal scale limitations of models for complex atomistic systems considered in biochemistry and materials science necessitates the development of enhanced sampling methods. The potential acceleration in exploring the configurational space by enhanced sampling methods depends on the choice of collective variables (CVs). In this work, we formulate the discovery of CVs as a Bayesian inference problem and consider the CVs as hidden generators of the full-atomistic trajectory. The ability to generate samples of the fine-scale atomistic configurations using limited training data allows us to compute estimates of observables as well as our probabilistic confidence on them. The methodology is based on emerging methodological advances in machine learning and variational inference. The discovered CVs are related to physicochemical properties which are essential for understanding mechanisms especially in unexplored complex systems. We provide a quantitative assessment of the CVs in terms of their predictive ability for alanine dipeptide (ALA-2) and ALA-15 peptide

Predictive Coarse-Graining

We propose a data-driven, coarse-graining formulation in the context of equilibrium statistical mechanics. In contrast to existing techniques which are based on a fine-to-coarse map, we adopt the opposite strategy by prescribing a probabilistic coarse-to-fine map. This corresponds to a directed probabilistic model where the coarse variables play the role of latent generators of the fine scale (all-atom) data. From an information-theoretic perspective, the framework proposed provides an improvement upon the relative entropy method and is capable of quantifying the uncertainty due to the information loss that unavoidably takes place during the CG process. Furthermore, it can be readily extended to a fully Bayesian model where various sources of uncertainties are reflected in the posterior of the model parameters. The latter can be used to produce not only point estimates of fine-scale reconstructions or macroscopic observables, but more importantly, predictive posterior distributions on these quantities. Predictive posterior distributions reflect the confidence of the model as a function of the amount of data and the level of coarse-graining. The issues of model complexity and model selection are seamlessly addressed by employing a hierarchical prior that favors the discovery of sparse solutions, revealing the most prominent features in the coarse-grained model. A flexible and parallelizable Monte Carlo - Expectation-Maximization (MC-EM) scheme is proposed for carrying out inference and learning tasks. A comparative assessment of the proposed methodology is presented for a lattice spin system and the SPC/E water model

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 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 benchmark problems including flow in heterogeneous media defined in terms of limited data-driven permeability realizations. 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 $4,225$ 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.Comment: 52 pages, 28 figures, submitted to Journal of Computational Physic

Structured Bayesian Gaussian process latent variable model

We introduce a Bayesian Gaussian process latent variable model that explicitly captures spatial correlations in data using a parameterized spatial kernel and leveraging structure-exploiting algebra on the model covariance matrices for computational tractability. Inference is made tractable through a collapsed variational bound with similar computational complexity to that of the traditional Bayesian GP-LVM. Inference over partially-observed test cases is achieved by optimizing a "partially-collapsed" bound. Modeling high-dimensional time series systems is enabled through use of a dynamical GP latent variable prior. Examples imputing missing data on images and super-resolution imputation of missing video frames demonstrate the model

Variational Reformulation of Bayesian Inverse Problems

The classical approach to inverse problems is based on the optimization of a misfit function. Despite its computational appeal, such an approach suffers from many shortcomings, e.g., non-uniqueness of solutions, modeling prior knowledge, etc. The Bayesian formalism to inverse problems avoids most of the difficulties encountered by the optimization approach, albeit at an increased computational cost. In this work, we use information theoretic arguments to cast the Bayesian inference problem in terms of an optimization problem. The resulting scheme combines the theoretical soundness of fully Bayesian inference with the computational efficiency of a simple optimization

Solving inverse problems using conditional invertible neural networks

Inverse modeling for computing a high-dimensional spatially-varying property field from indirect sparse and noisy observations is a challenging problem. This is due to the complex physical system of interest often expressed in the form of multiscale PDEs, the high-dimensionality of the spatial property of interest, and the incomplete and noisy nature of observations. To address these challenges, we develop a model that maps the given observations to the unknown input field in the form of a surrogate model. This inverse surrogate model will then allow us to estimate the unknown input field for any given sparse and noisy output observations. Here, the inverse mapping is limited to a broad prior distribution of the input field with which the surrogate model is trained. In this work, we construct a two- and three-dimensional inverse surrogate models consisting of an invertible and a conditional neural network trained in an end-to-end fashion with limited training data. The invertible network is developed using a flow-based generative model. The developed inverse surrogate model is then applied for an inversion task of a multiphase flow problem where given the pressure and saturation observations the aim is to recover a high-dimensional non-Gaussian permeability field where the two facies consist of heterogeneous permeability and varying length-scales. For both the two- and three-dimensional surrogate models, the predicted sample realizations of the non-Gaussian permeability field are diverse with the predictive mean being close to the ground truth even when the model is trained with limited data.Comment: 58 pages, 20 figure

Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data

Surrogate modeling and uncertainty quantification tasks for PDE systems are most often considered as supervised learning problems where input and output data pairs are used for training. The construction of such emulators is by definition a small data problem which poses challenges to deep learning approaches that have been developed to operate in the big data regime. Even in cases where such models have been shown to have good predictive capability in high dimensions, they fail to address constraints in the data implied by the PDE model. This paper provides a methodology that incorporates the governing equations of the physical model in the loss/likelihood functions. The resulting physics-constrained, deep learning models are trained without any labeled data (e.g. employing only input data) and provide comparable predictive responses with data-driven models while obeying the constraints of the problem at hand. This work employs a convolutional encoder-decoder neural network approach as well as a conditional flow-based generative model for the solution of PDEs, surrogate model construction, and uncertainty quantification tasks. The methodology is posed as a minimization problem of the reverse Kullback-Leibler (KL) divergence between the model predictive density and the reference conditional density, where the later is defined as the Boltzmann-Gibbs distribution at a given inverse temperature with the underlying potential relating to the PDE system of interest. The generalization capability of these models to out-of-distribution input is considered. Quantification and interpretation of the predictive uncertainty is provided for a number of problems.Comment: 51 pages, 18 figures, submitted to Journal of Computational Physic