Inverse and big-data problems are widespread in computational sciences and engineering. Despite formidable advances in recent years on all frontiers, ranging from pure mathematics to computational sciences, significant challenges remain, especially when it comes to addressing data-driven problems. In inverse/learning problems, parameters are typically related to indirect measurements by a system of partial differential equations (PDEs) or a network, which could be highly nonlinear and non-convex. Available indirect data are often noisy, and subject to natural variation, while the unknown parameters of interest are high dimensional, or possibly infinite-dimensional in principle. Bayesian inference provides a systematic framework that rigorously that allows us to quantify the uncertainty in the inverse/learning problems, and to assess model validity and adequacy. Since the amount of data we wish to process is only going to increase for the foreseeable future, there is a critical need for effective algorithms that integrate data with simulations and learning approaches that are computation- and data-scalable. This minisymposium aims to attract researchers at the forefront of inverse and learning problems, data science, and data-intensive problems to present their latest work on computation- and data-scalable algorithms in inverse problems and learning.
14:00
Post-Processing of High-Dimensional Data
Hermann G. Matthies | TU Braunschweig | Germany
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Authors:
Mike Espig | Westsaechsische Hochschule Zwickau | Germany
Wolfgang Hackbusch | Max Planck Institute for Mathematics in the Sciences, Leipzig | Germany
Alexander Litvinenko | RWTH Aachen | Germany
Hermann G. Matthies | TU Braunschweig | Germany
Elmar Zander | TU Braunschweig | Germany
Scientific computations or measurements may result in huge volumes of data. Often these can be thought of as a function on a high-dimensional domain, and can be conceptually arranged in the format of a tensor of high degree
in some lossy compressed format. We look at some common post-processing tasks which are not obvious in the compressed format, as such huge data sets can not be stored in their entirety, and the value of an element is not readily accessible through simple look-up. Considered tasks are finding the location of maximum or minimum, or finding the indices of all elements in some interval --- i.e. level sets, the number of elements in such a level set, the probability of a level set, and the mean and variance of the total collection. The algorithms for this are fixed point iterations to find functions of the data, yielding the desired result. For this, the data is considered as an element of an associative
commutative algebra. Such an algebra is isomorphic to a commutative sub-algebra of the usual matrix algebra, hence we use of matrix algorithms to compute functions of the data with algebra operations. We allow the actual computational representation to be a lossy compression, and allow the algebra operations to be performed in
an approximate fashion, so as to maintain a high compression level. One example of such compression addressed explicitly is the representation of data as a tensor in the form of a low-rank representation.
14:30
Data Driven Governing Equations Recovery with Deep Neural Networks
Dongbin Xiu | Ohio State University | United States
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Authors:
Dongbin Xiu | Ohio State University | United States
Tong Qin | Ohio State University | United States
Kailiang Wu | Ohio State University | United States
We present effective numerical algorithms for recovering unknown governing differential equations from measurement data. Upon recasting the problem into a function approximation problem, we discuss several important aspects for accurate recovery/approximation. Most notably, we discuss the importance of using a large number of short bursts of trajectory data, rather than using data from a single long trajectory. We also present several recovery strategies using deep neural networks (DNNs), especially those based on reside network (ResNet). We then present an extensive set of numerical examples of both linear and nonlinear systems to demonstrate the properties and effectiveness of our equation recovery algorithms.
15:00
- NEW - A Bayesian Approach to Real-Time Dynamic Parameter Estimation Using PMU Measurement
Xiao Chen | Lawrence Livermore National Laboratory | United States
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Author:
Xiao Chen | Lawrence Livermore National Laboratory | United States
In this work, we develop a polynomial-chaos-expansion (PCE)-based approach for decentralized dynamic parameter estimation through Bayesian inference. Using this approach, the non-Gaussian distribution of the inverted parameters is obtained. Specifically, we first represent the decentralized generator model with the PCE-based surrogate. This surrogate allows us to efficiently evaluate the time-consuming dynamic solver at parameter values through Metropolis-Hastings (M-H)-based Markov Chain Monte Carlo (MCMC). Then, we propose a two-stage Hybrid Markov Chain Monte Carlo (MCMC) method to recover a posterior distribution of the decentralized generator model parameters. In the first stage, we use the gradient-enhanced Langevin MCMC algorithm to characterize an intermediate posterior parameter distribution. This algorithm is computationally scalable to the high-dimensional parameter space. Based on the intermediate posterior distribution, during the second stage, we use the adaptive MCMC algorithm to fine-tune the strong correlations between the parameters. Finally, the fully recovered posterior distribution is obtained. The simulation results show that the proposed PCE-based Hybrid MCMC algorithm can accurately and efficiently estimate the high-dimensional generator dynamic model parameters with full probabilistic distribution provided.
15:30
Bayesian Inverse Problems Using Dimensionality Reduction and Machine Learning
Sheroze Sheriffdeen | Oden Institute of Computational Engineering and Sciences, The University of Texas at Austin | United States
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Authors:
Sheroze Sheriffdeen | Oden Institute of Computational Engineering and Sciences, The University of Texas at Austin | United States
Tan Bui-Thanh | Oden Institute of Computational Engineering and Sciences, The University of Texas at Austin | United States
Given a PDE-constrained Bayesian inference problem, a deep learning framework is proposed to improve the efficiency of the many-query nature of the sampling process by incorporating a reduced-order model augmented by a data-driven deep learning error model. Reduced-order models are derived from high fidelity models using approaches such as proper orthogonal decomposition, simplifying physics assumptions, using coarser grids, and looser residual tolerances. A reduced-order model provides rapid and reasonably accurate solutions to new parameters of interest, and are typically formed using expensive higher-fidelity model solutions to find the reduced subspace. These approximate reduced-order models reduce computational time but they introduce additional uncertainty to the solution.
We statistically model the error of the reduced-order model by training a deep neural network using a data-driven approach involving solution vectors computed from forward solves of the reduced-order model and the high fidelity model. The deep neural network is trained during the offline phase and the error bounds can be improved online as new training data is observed.
Furthermore, using automatic differentiation of the neural network and the adjoint method on the reduced-order model, we obtain gradients of the model with respect to the parameters of interest which can be used to accelerate the sampling-based Bayesian inference problem.