Data Integration Methods for Bayesian Inverse Problems
Master's Thesis for an MA in Mathematics at Arizona State University titled "Data Integration Methods for Bayesian Inverse Problems."
97 MATHEMATICS AND COMPUTING↗
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Results for “inverse problems, Bayesian modeling, uncertainty quantification”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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A Bayesian Calibration Framework with Embedded Model Error for Model Diagnostics
We study the utility and performance of a Bayesian model error embedding construction in the context of molecular dynamics modeling of metallic alloys, where we embed model error terms in existing interatomic potential model parameters. To alleviate the computational burden of this approach, we propose a framework combining likelihood approximation and Gaussian process surrogates. Here we leverage sparse Gaussian process techniques to construct a hierarchy of increasingly accurate but more expensive surrogate models. This hierarchy is then exploited by multilevel Markov chain Monte Carlo methods to efficiently sample from the target posterior distribution. We illustrate the utility of this approach by calibrating an interatomic potential model for a family of gold-copper alloys. In particular, this case study highlights effective means for dealing with computational challenges with Bayesian model error embedding in large-scale physical models, and the utility of embedded model error for model diagnostics.
Bayesian inference↗
A Variational Inference Approach to Inverse Problems with Gamma Hyperpriors
Hierarchical models with gamma hyperpriors provide a flexible, sparse-promoting framework to bridge L 1 and L 2 regularizations in Bayesian formulations to inverse problems. Despite the Bayesian motivation for these models, existing methodologies are limited to maximum a posteriori estimation. The potential to perform uncertainty quantification has not yet been realized. This paper introduces a variational iterative alternating scheme for hierarchical inverse problems with gamma hyperpriors. The proposed variational inference approach yields accurate reconstruction, provides meaningful uncertainty quantification, and is easy to implement. In addition, it lends itself naturally to conduct model selection for the choice of hyperparameters. Here, we illustrate the performance of our methodology in several computed examples, including a deconvolution problem and sparse identification of dynamical systems from time series data.
Bayesian shrinkage↗
Thinking Bayesian for plasma physicists
Bayesian statistics offers a powerful technique for plasma physicists to infer knowledge from the heterogeneous data types encountered. To explain this power, a simple example, Gaussian Process Regression, and the application of Bayesian statistics to inverse problems are explained. The likelihood is the key distribution because it contains the data model, or theoretic predictions, of the desired quantities. By using prior knowledge, the distribution of the inferred quantities of interest based on the data given can be inferred. Because it is a distribution of inferred quantities given the data and not a single prediction, uncertainty quantification is a natural consequence of Bayesian statistics. The benefits of machine learning in developing surrogate models for solving inverse problems are discussed, as well as progress in quantitatively understanding the errors that such a model introduces.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Efficient Neural Network Approaches for Conditional Optimal Transport with Applications in Bayesian Inference
In this work, we present two neural network approaches that approximate the solutions of static and dynamic conditional optimal transport (COT) problems. Both approaches enable conditional sampling and conditional density estimation, which are core tasks in Bayesian inference—particularly in the simulation-based (“likelihood-free”) setting. Our methods represent the target conditional distribution as a transformation of a tractable reference distribution. Obtaining such a transformation, chosen here to be an approximation of the COT map, is computationally challenging even in moderate dimensions. To improve scalability, our numerical algorithms use neural networks to parameterize candidate maps and further exploit the structure of the COT problem. Our static approach approximates the map as the gradient of a partially input convex neural network. It uses a novel numerical implementation to increase computational efficiency compared to state-of-the-art alternatives. Our dynamic approach approximates the conditional optimal transport via the flow map of a regularized neural ODE; compared to the static approach, it is slower to train but offers more modeling choices and can lead to faster sampling. We demonstrate both algorithms numerically, comparing them with competing state-of-the-art approaches, using benchmark datasets and simulation-based Bayesian inverse problems.
97 MATHEMATICS AND COMPUTING↗
New approaches to Bayesian uncertainty quantification for Nuclear Science (Final Technical Report)
Inverse problems play a central role in experimentation and theory/data comparisons for many areas of modern Nuclear Physics (NP) and High-Energy Physics (HEP). Bayes’s Theorem is a powerful tool for solving Inverse Problems, providing conceptually transparent and unbiased constraints on theoretical parameters and their uncertainties (“Bayesian Inference”) and enabling the quantification of agreement or tension between models and data. However, analyses based on Bayesian Inference are often challenging for NP and HEP applications, either because of the large number of parameters in the problem, the high computational cost, or both. We propose a multi-institutional collaboration to develop and deploy novel Bayesian analysis tools that advance the scientific scope of a broad range of current and future NP experiments. This project brings together NP domain scientists working on several high-profile NP projects for which new, high-performance Bayesian Uncertainty Quantification (“Bayesian UQ”) methods are essential to carry out the science, and data scientists who are developing state-of-the-art methods applicable to these problems. The NP projects in this proposal comprise measurements of the mass and fundamental nature of the neutrino; study of the Quark-Gluon Plasma that filled the early universe; and mapping of natural and anthropogenic radiation environments. While these NP projects have very different scientific goals, with datasets and analysis approaches that differ significantly, they share common requirements for improving computationally intensive Bayesian analyses using advanced Machine Learning algorithms and will benefit strongly from a coherent effort to develop general solutions. This proposal brings together these projects and forefront ML-based data science algorithms to develop such general solutions. The methods developed in this project will also be more widely applicable, thereby advancing science in the larger Nuclear Physics portfolio.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Solving Stochastic Inverse Problems for Property–Structure Linkages Using Data-Consistent Inversion and Machine Learning
Determining process–structure–property linkages is one of the key objectives in material science, and uncertainty quantification plays a critical role in understanding both process–structure and structure–property linkages. In this work, we seek to learn a distribution of microstructure parameters that are consistent in the sense that the forward propagation of this distribution through a crystal plasticity finite element model matches a target distribution on materials properties. This stochastic inversion formulation infers a distribution of acceptable/consistent microstructures, as opposed to a deterministic solution, which expands the range of feasible designs in a probabilistic manner. Furthermore, to solve this stochastic inverse problem, we employ a recently developed uncertainty quantification framework based on push-forward probability measures, which combines techniques from measure theory and Bayes’ rule to define a unique and numerically stable solution. This approach requires making an initial prediction using an initial guess for the distribution on model inputs and solving a stochastic forward problem. To reduce the computational burden in solving both stochastic forward and stochastic inverse problems, we combine this approach with a machine learning Bayesian regression model based on Gaussian processes and demonstrate the proposed methodology on two representative case studies in structure–property linkages.
36 MATERIALS SCIENCE↗
Desmearing Bonse–Hart USANS data using Bayesian Gaussian process regression
Ultra-small-angle neutron scattering (USANS) enables access to micrometer-scale structures but is intrinsically affected by strong, anisotropic resolution smearing arising from slit-geometry optics. As a result, recovery of the intrinsic scattering intensity constitutes an ill-posed inverse problem, and commonly used iterative desmearing methods lack rigorous uncertainty quantification. We present a Bayesian desmearing framework for slit-geometry USANS based on Gaussian process regression. In this approach, the scattering intensity is modeled as a smooth random function, and the instrumental point spread function is incorporated explicitly as a forward operator. The resulting formulation yields a closed-form maximum a posteriori solution with well-defined credibility intervals. Computational benchmarks and experimental validation using combined USANS and small-angle neutron scattering (SANS) measurements demonstrate that the framework enables stable desmearing, suppresses experimental noise, and preserves physically meaningful structural features under realistic conditions.
Tung, Chi-Huan [Oak Ridge National Laboratory (ORN↗
Bayesian inverse uncertainty quantification of the physical model parameters for the spallation neutron source first target station
Not Available
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
B-PINNs: Bayesian Physics-informal Neural Networks for Forward and Inverse PDE Problems with Noisy Data
We propose a Bayesian physics-informed neural network (B-PINN) to solve both forward and inverse nonlinear problems described by partial differential equations (PDEs) and noisy data. In this Bayesian framework, the Bayesian neural network (BNN) combined with a PINN for PDEs serves as the prior while the Hamiltonian Monte Carlo (HMC) or the variational inference (VI) could serve as an estimator of the posterior. B-PINNs make use of both physical laws and scattered noisy measurements to provide predictions and quantify the aleatoric uncertainty arising from the noisy data in the Bayesian framework. Compared with PINNs, in addition to uncertainty quantification, B-PINNs obtain more accurate predictions in scenarios with large noise due to their capability of avoiding overfitting. We conduct a systematic comparison between the two different approaches for the B-PINNs posterior estimation (i.e., HMC or VI), along with dropout used for quantifying uncertainty in deep neural networks. Our experiments show that HMC is more suitable than VI with mean field Gaussian approximation for the B-PINNs posterior estimation, while dropout employed in PINNs can hardly provide accurate predictions with reasonable uncertainty. Finally, we replace the BNN in the prior with a truncated Karhunen-Loève (KL) expansion combined with HMC or a deep normalizing flow (DNF) model as posterior estimators. The KL is as accurate as BNN and much faster but this framework cannot be easily extended to high-dimensional problems unlike the BNN based framework.
Non linear PDEs, Noisy data, Bysian physics inform↗
Multi-Modal Bayesian Neural Network Surrogates with Conjugate Last-Layer Estimation
As data collection and simulation capabilities advance, multi-modal learning, the task of learning from multiple modalities and sources of data, is becoming an increasingly important area of research. Surrogate models that learn from data of multiple auxiliary modalities to support the modeling of a highly expensive quantity of interest have the potential to aid outer loop applications such as optimization, inverse problems, or sensitivity analyses when multi-modal data are available. We develop two multi-modal Bayesian neural network surrogate models and leverage conditionally conjugate distributions in the last layer to estimate model parameters using stochastic variational inference (SVI). We provide a method to perform this conjugate SVI estimation in the presence of partially missing observations. Here, we demonstrate improved prediction accuracy and uncertainty quantification compared to unimodal surrogate models for both scalar and time series data.
97 MATHEMATICS AND COMPUTING↗
Bayesian Poroelastic Aquifer Characterization From InSAR Surface Deformation Data. 2. Quantifying the Uncertainty
Uncertainty quantification of groundwater (GW) aquifer parameters is critical for efficient management and sustainable extraction of GW resources. These uncertainties are introduced by the data, model, and prior information on the parameters. We develop a Bayesian inversion framework that uses Interferometric Synthetic Aperture Radar (InSAR) surface deformation data to infer the laterally heterogeneous permeability of a transient linear poroelastic model of a confined GW aquifer. The Bayesian solution of this inverse problem takes the form of a posterior probability density of the permeability. Exploring this posterior using classical Markov chain Monte Carlo (MCMC) methods is computationally prohibitive due to the large dimension of the discretized permeability field and the expense of solving the poroelastic forward problem. However, in many partial differential equation (PDE)-based Bayesian inversion problems, the data are only informative in a few directions in parameter space. For the poroelasticity problem, we prove this property theoretically for a one-dimensional problem and demonstrate it numerically for a three-dimensional aquifer model. Here we design a generalized preconditioned Crank-Nicolson (gpCN) MCMC method that exploits this intrinsic low dimensionality by using a low-rank-based Laplace approximation of the posterior as a proposal, which we build scalably. The feasibility of our approach is demonstrated through a real GW aquifer test in Nevada. The inherently two-dimensional nature of InSAR surface deformation data informs a sufficient number of modes of the permeability field to allow detection of major structures within the aquifer, significantly reducing the uncertainty in the pressure and the displacement quantities of interest.
54 ENVIRONMENTAL SCIENCES↗
Variational encoder geostatistical analysis (VEGAS) with an application to large scale riverine bathymetry
Estimation of riverbed profiles, also known as bathymetry, plays a vital role in many applications, such as safe and efficient inland navigation, prediction of bank erosion, land subsidence, and flood risk management. The high cost and complex logistics of direct bathymetry surveys, i.e, depth imaging, have encouraged the use of indirect measurements such as surface flow velocities. However, estimating high-resolution bathymetry from indirect measurements is an inverse problem that can be computationally challenging. Here, we propose a reduced-order model (ROM) based approach that utilizes a variational autoencoder (VAE), a type of deep neural network with a narrow layer in the middle, to compress bathymetry and flow velocity information and accelerate bathymetry inverse problems from flow velocity measurements. In our application, the shallow-water equations (SWE) with appropriate boundary conditions (BCs), e.g., the discharge and/or the free surface elevation, constitute the forward problem, to predict flow velocity. Then, ROMs of the SWEs are constructed on a nonlinear manifold of low dimensionality through a variational encoder and the bathymetry inversion problem is derived on the low-dimensional latent space in a Hierarchical Bayesian setting. Further, the reformulation allows variational inference with a small number (e.g., $\mathscr{O}$ (100) of ROM runs and efficient uncertainty quantification. We have tested our inversion approach on a one-mile reach of the Savannah River, GA, USA. Once the neural network is trained (offline stage), the proposed technique can perform the inversion operation orders of magnitude faster than traditional inversion methods that are commonly based on linear projections, such as principal component analysis (PCA), or the principal component geostatistical approach (PCGA). Furthermore, tests show that the algorithm can estimate the bathymetry with good accuracy even with sparse flow velocity measurements.
54 ENVIRONMENTAL SCIENCES↗
Bayesian Inference and the Effects of Varying Uncertainty Models in Charring Ablator Calibration and Uncertainty Quantification Problems
The Mars Science Laboratory (MSL) vehicle utilized a heat shield constructed from NASA’s Phenolic-Impregnated Carbon Ablator (PICA) material to protect the main structure from the high enthalpy environment encountered during hypersonic atmospheric entry. During the vehicle’s descent through Martian atmosphere, multiple thermocouples embedded within the heat shield captured in-depth material temperature data that allow for studies to be conducted on current material response reconstruction tools. In the present work, material temperature data obtained from thermocouples within the MISP-4 plug (MEDLI (Mars Science Laboratory Entry, Descent, and Landing Instrument) Integrated Sensor Plug) are utilized in the calibration of Theoretical Ablative Composite for Open Testing (TACOT) model parameters in conjunction with NASA’s Porous material Analysis Toolbox (PATO) through Bayesian inference where uncertainty due to parametric, modeling, and experimental sources is simultaneously quantified. Prior to the study, a sensitivity analysis is performed through computation of the robust Sobol indices in an effort to study the relationship between input space and model response and to reduce the dimensionality of the statistical inverse problem. The Bayesian inference methodology necessitates an a-priori choice to be made for the uncertainty model for which numerous possibilities are available. Across most works, however, only basic additive or multiplicative models are utilized with pre-defined magnitudes of uncertainty based on a-priori knowledge or to-be-calibrated multipliers of static covariance matrix structures. The present effort explores the effects of informed uncertainty models, ones with temporal dependence that are simultaneously calibrated through Bayesian inference, on calibrated results for parameters that make up the uncertain input space.
Sensitivity Analysis↗
Statistical modelling and Bayesian inversion for a Compton imaging system: application to radioactive source localization
Abstract This paper presents a statistical forward model for a Compton imaging system, called Compton imager. This system, under development at the University of Illinois Urbana Champaign, is a variant of Compton cameras with a single type of sensors which can simultaneously act as scatterers and absorbers. This imager is convenient for imaging situations requiring a wide field of view. The proposed statistical forward model is then used to solve the inverse problem of estimating the location and energy of point-like sources from observed data. This inverse problem is formulated and solved in a Bayesian framework by using a Metropolis within Gibbs algorithm for the estimation of the location, and an expectation-maximization algorithm for the estimation of the energy. This approach leads to more accurate estimation when compared with the deterministic standard back-projection approach, with the additional benefit of uncertainty quantification in the low photon imaging setting.
Tarpau, Cécilia (ORCID:0000000286539490)↗
Robust scalable initialization for Bayesian variational inference with multi-modal Laplace approximations
Predictive modeling typically relies on Bayesian model calibration to provide uncertainty quantification. Variational inference utilizing fully independent (“mean-field”) Gaussian distributions are often used as approximate probability density functions. This simplification is attractive since the number of variational parameters grows only linearly with the number of unknown model parameters. However, the resulting diagonal covariance structure and unimodal behavior can be too restrictive to provide useful approximations of intractable Bayesian posteriors that exhibit highly non-Gaussian behavior, including multimodality. High-fidelity surrogate posteriors for these problems can be obtained by considering the family of Gaussian mixtures. Gaussian mixtures are capable of capturing multiple modes and approximating any distribution to an arbitrary degree of accuracy, while maintaining some analytical tractability. Unfortunately, variational inference using Gaussian mixtures with full-covariance structures suffers from a quadratic growth in variational parameters with the number of model parameters. The existence of multiple local minima due to strong nonconvex trends in the loss functions often associated with variational inference present additional complications, These challenges motivate the need for robust initialization procedures to improve the performance and computational scalability of variational inference with mixture models. In this work, we propose a method for constructing an initial Gaussian mixture model approximation that can be used to warm-start the iterative solvers for variational inference. The procedure begins with a global optimization stage in model parameter space. In this step, local gradient-based optimization, globalized through multistart, is used to determine a set of local maxima, which we take to approximate the mixture component centers. Around each mode, a local Gaussian approximation is constructed via the Laplace approximation. Finally, the mixture weights are determined through constrained least squares regression. The robustness and scalability of the proposed methodology is demonstrated through application to an ensemble of synthetic tests using high-dimensional, multimodal probability density functions. Here, the practical aspects of the approach are demonstrated with inversion problems in structural dynamics.
97 MATHEMATICS AND COMPUTING↗
Goal-oriented real-time Bayesian inference for linear autonomous dynamical systems with application to digital twins for tsunami early warning
We present a goal-oriented framework for constructing digital twins with the following properties: (1) they employ discretizations of high-fidelity partial differential equation (PDE) models governed by autonomous dynamical systems, leading to large-scale forward problems; (2) they solve a linear inverse problem to assimilate observational data to infer uncertain model components followed by a forward prediction of the evolving dynamics; and (3) the entire end-to-end, data-to-inference-to-prediction computation is carried out without approximation and in real time through a Bayesian framework that rigorously accounts for uncertainties. Several challenges must be overcome to realize this framework, including the large scale of the forward problem, the high dimensionality of the parameter space, and for a class of problems including those we target, the slow decay of the singular values of the parameter-to-observable map. Here we introduce a methodology to overcome these challenges by exploiting the autonomous structure of the forward model to decompose the solution of the inverse problem into a one-time-only offline phase in which the PDE model is solved a limited number of times (equal to the number of sensors), and an online phase that maps well onto GPUs and computes the parameter inference and prediction of quantities of interest in real time, given observational data. Our ultimate goal is to apply this framework to construct digital twins for subduction zones, including Cascadia, to provide early warning for tsunamis generated by megathrust earthquakes. To this end, we demonstrate how our methodology can be used to employ seafloor pressure observations, along with the coupled acoustic–gravity wave equations, to infer the earthquake-induced spatiotemporal seafloor motion (discretized with $\mathscr{O}$ (10 9 ) parameters) and forward predict the tsunami propagation. We present results of an end-to-end inference, prediction, and uncertainty quantification for a representative test problem with $\mathscr{O}$ (10 8 ) inversion parameters for which goal-oriented Bayesian inference is accomplished exactly and in real time, that is, in a matter of seconds.
97 MATHEMATICS AND COMPUTING↗
A mixed, unified forward/inverse framework for earthquake problems: fault implementation and coseismic slip estimate
SUMMARY We introduce a new finite-element (FE) based computational framework to solve forward and inverse elastic deformation problems for earthquake faulting via the adjoint method. Based on two advanced computational libraries, FEniCS and hIPPYlib for the forward and inverse problems, respectively, this framework is flexible, transparent and easily extensible. We represent a fault discontinuity through a mixed FE elasticity formulation, which approximates the stress with higher order accuracy and exposes the prescribed slip explicitly in the variational form without using conventional split node and decomposition discrete approaches. This also allows the first order optimality condition, that is the vanishing of the gradient, to be expressed in continuous form, which leads to consistent discretizations of all field variables, including the slip. We show comparisons with the standard, pure displacement formulation and a model containing an in-plane mode II crack, whose slip is prescribed via the split node technique. We demonstrate the potential of this new computational framework by performing a linear coseismic slip inversion through adjoint-based optimization methods, without requiring computation of elastic Green’s functions. Specifically, we consider a penalized least squares formulation, which in a Bayesian setting—under the assumption of Gaussian noise and prior—reflects the negative log of the posterior distribution. The comparison of the inversion results with a standard, linear inverse theory approach based on Okada’s solutions shows analogous results. Preliminary uncertainties are estimated via eigenvalue analysis of the Hessian of the penalized least squares objective function. Our implementation is fully open-source and Jupyter notebooks to reproduce our results are provided. The extension to a fully Bayesian framework for detailed uncertainty quantification and non-linear inversions, including for heterogeneous media earthquake problems, will be analysed in a forthcoming paper.
58 GEOSCIENCES↗