We investigate a deep learning approach to efficiently perform Bayesian inference in partial differential equation (PDE) and integral equation models over potentially high-dimensional parameter spaces. The contributions of this paper are two-fold; the first is the introduction of a neural network approach to approximating the solutions of Fredholm and Volterra integral equations of the first and second kind. The second is the development of a new, efficient deep learning-based method for Bayesian inversion applied to problems that can be described by PDEs or integral equations. To achieve this we introduce a surrogate model, and demonstrate how this allows efficient sampling from a Bayesian posterior distribution in which the likelihood depends on the solutions of PDEs or integral equations. Our method relies on the direct approximation of parametric solutions by neural networks, without need of traditional numerical solves. This deep learning approach allows the accurate and efficient approximation of parametric solutions in significantly higher dimensions than is possible using classical discretisation schemes. Since the approximated solutions can be cheaply evaluated, the solutions of Bayesian inverse problems over large parameter spaces are efficient using Markov chain Monte Carlo. We demonstrate the performance of our method using two real-world examples; these include Bayesian inference in the PDE and integral equation case for an example from electrochemistry, and Bayesian inference of a function-valued heat-transfer parameter with applications in aviation.
SURGIN: SURrogate-guided Generative INversion for subsurface multiphase flow with quantified uncertainty
Department of Geotechnical Engineering, College of Civil Engineering, Tongji University（同济大学土木工程学院地质工程系）；Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University（教育部同济大学地质与地下工程重点实验室）；Physical Science and Engineering Division, King Abdullah University of Science and Technology (KAUST)（卡布斯科学与技术大学（KAUST）物理科学与工程系）；State Key Laboratory of Marine Geology, Tongji University（同济大学海洋地质国家重点实验室）；Tencent SSV（腾讯SSV）；Tencent SD（腾讯SD）
Gaussian process surrogate with physical law-corrected prior for multi-coupled PDEs defined on irregular geometry
School of Artificial Intelligence, Wuhan University（武汉大学人工智能学院）；School of Mathematics and Statistics, Central South University（中南大学数学与统计学院）；Institute of Applied Physics and Computational Mathematics（北京应用物理与计算数学研究所）；Institute of Mathematics, Henan Academy of Sciences（河南省科学院数学研究所）
University of New South Wales（新南威尔士大学）；University of Sydney（悉尼大学）；SRM Institute of Science and Technology（SRM科学技术研究所）
The complex and computationally expensive nature of landscape evolution models pose significant challenges in the inference and optimisation of unknown parameters. Bayesian inference provides a methodology for estimation and uncertainty quantification of unknown model parameters. In our previous work, we developed parallel tempering Bayeslands as a framework for parameter estimation and uncertainty quantification for the Badlands landscape evolution model. Parallel tempering Bayeslands features high-performance computing with dozens of processing cores running in parallel to enhance computatio…
We present an extensible software framework, hIPPYlib, for solution of large-scale deterministic and Bayesian inverse problems governed by partial differential equations (PDEs) with infinite-dimensional parameter fields (which are high-dimensional after discretization). hIPPYlib overcomes the prohibitive nature of Bayesian inversion for this class of problems by implementing state-of-the-art scalable algorithms for PDE-based inverse problems that exploit the structure of the underlying operators, notably the Hessian of the log-posterior. The key property of the algorithms implemented in hIPPYl…
University of Warwick（华威大学）；The Alan Turing Institute（艾伦·图灵研究所）
Specifying a governing physical model in the presence of missing physics and recovering its parameters are two intertwined and fundamental problems in science. Modern machine learning allows one to circumvent these, via emulators and surrogates, but in doing so disregards prior knowledge and physical laws that are especially important for small data regimes, interpretability, and decision making. In this work we fold the mechanistic model into a flexible data-driven surrogate to arrive at a physically structured decoder network. This provides accelerated inference for the Bayesian inverse prob…
Bayesian inverse modeling is important for a better understanding of hydrological processes. However, this approach can be computationally demanding, as it usually requires a large number of model evaluations. To address this issue, one can take advantage of surrogate modeling techniques. Nevertheless, when approximation error of the surrogate model is neglected, the inversion result will be biased. In this paper, we develop a surrogate-based Bayesian inversion framework that explicitly quantifies and gradually reduces the approximation error of the surrogate. Specifically, two strategies are …
This study introduces a non-intrusive approach in the context of low-rank separated representation to construct a surrogate of high-dimensional stochastic functions, e.g., PDEs/ODEs, in order to decrease the computational cost of Markov Chain Monte Carlo simulations in Bayesian inference. The surrogate model is constructed via a regularized alternative least-square regression with Tikhonov regularization using a roughening matrix computing the gradient of the solution, in conjunction with a perturbation-based error indicator to detect optimal model complexities. The model approximates a vector…
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…
We present an application of deep generative models in the context of partial-differential equation (PDE) constrained inverse problems. We combine a generative adversarial network (GAN) representing an a priori model that creates subsurface geological structures and their petrophysical properties, with the numerical solution of the PDE governing the propagation of acoustic waves within the earth's interior. We perform Bayesian inversion using an approximate Metropolis-adjusted Langevin algorithm (MALA) to sample from the posterior given seismic observations. Gradients with respect to the model…
A novel refinement measure for non-intrusive surrogate modelling of partial differential equations (PDEs) with uncertain parameters is proposed. Our approach uses an empirical interpolation procedure, where the proposed refinement measure is based on a PDE residual and probability density function of the uncertain parameters, and excludes parts of the PDE solution that are not used to compute the quantity of interest. The PDE residual used in the refinement measure is computed by using all the partial derivatives that enter the PDE separately. The proposed refinement measure is suited for effi…
SINTEF（挪威科学与工业研究基金会）；ETH Zürich（苏黎世联邦理工学院）；University of Southern California（南加州大学）；TIFR Centre for Applicable Mathematics（塔塔基础研究院应用数学中心）
We present a novel active learning algorithm, termed as iterative surrogate model optimization (ISMO), for robust and efficient numerical approximation of PDE constrained optimization problems. This algorithm is based on deep neural networks and its key feature is the iterative selection of training data through a feedback loop between deep neural networks and any underlying standard optimization algorithm. Under suitable hypotheses, we show that the resulting optimizers converge exponentially fast (and with exponentially decaying variance), with respect to increasing number of training sample…
In this contribution, we develop an efficient surrogate modeling framework for simulation-based optimization of enhanced oil recovery, where we particularly focus on polymer flooding. The computational approach is based on an adaptive training procedure of a neural network that directly approximates an input-output map of the underlying PDE-constrained optimization problem. The training process thereby focuses on the construction of an accurate surrogate model solely related to the optimization path of an outer iterative optimization loop. True evaluations of the objective function are used to…
College of Computing, Georgia Institute of Technology（佐治亚理工学院计算学院）；Department of Mathematics and Systems Engineering, Florida Institute of Technology（佛罗里达理工学院数学与系统工程系）
Advanced Computing, Mathematics and Data Division, Pacific Northwest National Laboratory（太平洋西北国家实验室先进计算、数学和数据 division）；Pacific Northwest National Laboratory（太平洋西北国家实验室）；Atmospheric Sciences and Global Change Division, Pacific Northwest National Laboratory（太平洋西北国家实验室大气科学与全球变化 division）
This paper presents a novel numerical method for the Newton seismic full-waveform inversion (FWI). The method is based on the full-space approach, where the state, adjoint state, and control variables are optimized simultaneously. Each Newton step is formulated as a PDE-constrained optimization problem, which is cast in the form of the Karush-Kuhn-Tucker (KKT) system of linear algebraic equitations. The KKT system is solved inexactly with a preconditioned Krylov solver. We introduced two preconditioners: the one based on the block-triangular factorization and its variant with an inexact block …
Sparse-grids-like surrogate models enhanced with gradient information
Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes”, Consiglio Nazionale delle Ricerche（应用数学与信息技术研究所“E.马格内斯”，意大利国家研究委员会）；Institute for Complex Molecular Systems, Department of Biomedical Engineering, Eindhoven University of Technology（复杂分子系统研究所，生物医学工程系，埃因霍温理工大学）
Mathematical modeling and simulation of complex physical systems based on partial differential equations (PDEs) have been widely used in engineering and industrial applications. To enable reliable predictions, it is crucial yet challenging to calibrate the model by inferring unknown parameters/fields (e.g., boundary conditions, mechanical properties, and operating parameters) from sparse and noisy measurements, which is known as a PDE-constrained inverse problem. In this work, we develop a novel bi-fidelity (BF) ensemble Kalman inversion method to tackle this challenge, leveraging the accuracy…
We present a new approach for constructing a data-driven surrogate model and using it for Bayesian parameter estimation in partial differential equation (PDE) models. We first use parameter observations and Gaussian Process regression to condition the Karhunen-Loéve (KL) expansion of the unknown space-dependent parameters and then build the conditional generalized Polynomial Chaos (gPC) surrogate model of the PDE states. Next, we estimate the unknown parameters by computing coefficients in the KL expansion minimizing the square difference between the gPC predictions and measurements of the sta…
In this paper, we develop an asymptotic expansion-regularization (AER) method for inverse source problems in two-dimensional nonlinear and nonstationary singularly perturbed partial differential equations (PDEs). The key idea of this approach is the use of the asymptotic-expansion theory, which allows us to determine the conditions for the existence and uniqueness of a solution to a given PDE with a sharp transition layer. As a by-product, we derive a simpler link equation between the source function and first-order asymptotic approximation of the measurable quantities, and based on that equat…
