Browsing by Author "Popov, Andrey A."
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- Adjoint-Matching Neural Network Surrogates for Fast 4D-Var Data AssimilationChennault, Austin; Popov, Andrey A.; Subrahmanya, Amit N.; Cooper, Rachel; Karpatne, Anuj; Sandu, Adrian (2021-11-16)The data assimilation procedures used in many operational numerical weather forecasting systems are based around variants of the 4D-Var algorithm. The cost of solving the 4D-Var problem is dominated by the cost of forward and adjoint evaluations of the physical model. This motivates their substitution by fast, approximate surrogate models. Neural networks offer a promising approach for the data-driven creation of surrogate models. The accuracy of the surrogate 4D-Var problem’s solution has been shown to depend explicitly on accurate modeling of the forward and adjoint for other surrogate modeling approaches and in the general nonlinear setting. We formulate and analyze several approaches to incorporating derivative information into the construction of neural network surrogates. The resulting networks are tested on out of training set data and in a sequential data assimilation setting on the Lorenz-63 system. Two methods demonstrate superior performance when compared with a surrogate network trained without adjoint information, showing the benefit of incorporating adjoint information into the training process.
- A Bayesian approach to multivariate adaptive localization in ensemble-based data assimilation with time-dependent extensionsPopov, Andrey A.; Sandu, Adrian (Copernicus Publications, 2019-06-14)Ever since its inception, the ensemble Kalman filter (EnKF) has elicited many heuristic approaches that sought to improve it. One such method is covariance localization, which alleviates spurious correlations due to finite ensemble sizes by using relevant spatial correlation information. Adaptive localization techniques account for how correlations change in time and space, in order to obtain improved covariance estimates. This work develops a Bayesian approach to adaptive Schur-product localization for the deterministic ensemble Kalman filter (DEnKF) and extends it to support multiple radii of influence. We test the proposed adaptive localization using the toy Lorenz’96 problem and a more realistic 1.5-layer quasi-geostrophic model. Results with the toy problem show that the multivariate approach informs us that strongly observed variables can tolerate larger localization radii. The univariate approach leads to markedly improved filter performance for the realistic geophysical model, with a reduction in error by as much as 33 %.
- An Ensemble Variational Fokker-Planck Method for Data AssimilationSubrahmanya, Amit N.; Popov, Andrey A.; Sandu, Adrian (2021-11-27)Particle flow filters that aim to smoothly transform particles from samples from a prior distribution to samples from a posterior are a major topic of active research. In this work we introduce a generalized framework which we call the the Variational Fokker-Planck method for filtering and smoothing in data assimilation that encompasses previous methods such as the mapping particle filter and the particle flow filter. By making use of the properties of the optimal Ito process that solves the underlying Fokker-Planck equation we can explicitly include heuristics methods such as rejuvenation and regularization that fit into this framework. We also extend our framework to higher dimensions using localization and covariance shrinkage, and provide a robust implicit-explicit method for solving the stochastic initial value problem describing the Ito process. The effectiveness of the variational Fokker-Planck method is demonstrated on three test problems, namely the Lorenz '63, Lorenz '96 and the quasi-geostrophic equations.
- A Fast Time-Stepping Strategy for Dynamical Systems Equipped With a Surrogate ModelRoberts, Steven; Popov, Andrey A.; Sarshar, Arash; Sandu, Adrian (Society for Industrial & Applied Mathematics (SIAM), 2022-01-01)Simulation of complex dynamical systems arising in many applications is computationally challenging due to their size and complexity. Model order reduction, machine learning, and other types of surrogate modeling techniques offer cheaper and simpler ways to describe the dynamics of these systems but are inexact and introduce additional approximation errors. In order to overcome the computational difficulties of the full complex models, on one hand, and the limitations of surrogate models, on the other, this work proposes a new accelerated time-stepping strategy that combines information from both. This approach is based on the multirate infinitesimal general-structure additive Runge–Kutta framework. The inexpensive surrogate model is integrated with a small time step to guide the solution trajectory, and the full model is treated with a large time step to occasionally correct for the surrogate model error and ensure convergence. We provide a theoretical error analysis, and several numerical experiments, to show that this approach can be significantly more efficient than using only the full or only the surrogate model for the integration.
- Investigation of Nonlinear Model Order Reduction of the Quasigeostrophic Equations through a Physics-Informed Convolutional AutoencoderCooper, Rachel; Popov, Andrey A.; Sandu, Adrian (2021-08-27)Reduced order modeling (ROM) is a field of techniques that approximates complex physics-based models of real-world processes by inexpensive surrogates that capture important dynamical characteristics with a smaller number of degrees of freedom. Traditional ROM techniques such as proper orthogonal decomposition (POD) focus on linear projections of the dynamics onto a set of spectral features. In this paper we explore the construction of ROM using autoencoders (AE) that perform nonlinear projections of the system dynamics onto a low dimensional manifold learned from data. The approach uses convolutional neural networks (CNN) to learn spatial features as opposed to spectral, and utilize a physics informed (PI) cost function in order to capture temporal features as well. Our investigation using the quasi-geostrophic equations reveals that while the PI cost function helps with spatial reconstruction, spatial features are less powerful than spectral features, and that construction of ROMs through machine learning-based methods requires significant investigation into novel non-standard methodologies.
- Modeling Of Fiber-Optic Sensors Based on Micromechanical Vibrations in LiquidProkhorov, A. M.; Claus, Richard O.; Popov, Andrey A.; Tulaikova, T. V. (Optical Society of America, 1997-08-01)Fiber-optic chemical sensors based on optical power absorption or wavelength changes are well known. A new type of sensing element is considered. A micromechanical vibrated tiber-optic tip changes its resonance frequency during its operation. Sensors of this type are simple and convenient and do not require adjustment while in use. They are useful in industry and in medical applications. The action of this sensitive element in a liquid is considered. (C) 1997 Optical Society of America.
- Multifidelity Ensemble Kalman Filtering Using Surrogate Models Defined by Theory-Guided AutoencodersPopov, Andrey A.; Sandu, Adrian (Frontiers, 2022-06-02)Data assimilation is a Bayesian inference process that obtains an enhanced understanding of a physical system of interest by fusing information from an inexact physics-based model, and from noisy sparse observations of reality. The multifidelity ensemble Kalman filter (MFEnKF) recently developed by the authors combines a full-order physical model and a hierarchy of reduced order surrogate models in order to increase the computational efficiency of data assimilation. The standard MFEnKF uses linear couplings between models, and is statistically optimal in case of Gaussian probability densities. This work extends the MFEnKF into to make use of a broader class of surrogate model such as those based on machine learning methods such as autoencoders non-linear couplings in between the model hierarchies. We identify the right-invertibility property for autoencoders as being a key predictor of success in the forecasting power of autoencoder-based reduced order models. We propose a methodology that allows us to construct reduced order surrogate models that are more accurate than the ones obtained via conventional linear methods. Numerical experiments with the canonical Lorenz'96 model illustrate that nonlinear surrogates perform better than linear projection-based ones in the context of multifidelity ensemble Kalman filtering. We additionality show a large-scale proof-of-concept result with the quasi-geostrophic equations, showing the competitiveness of the method with a traditional reduced order model-based MFEnKF.
- Physics-informed neural networks for PDE-constrained optimization and controlBarry-Straume, Jostein; Sarshar, Arash; Popov, Andrey A.; Sandu, Adrian (2022-05-06)A fundamental problem of science is designing optimal control policies that manipulate a given environment into producing the desired outcome. Control PhysicsInformed Neural Networks simultaneously solve a given system state, and its respective optimal control, in a one-stage framework that conforms to the physical laws of the system. Prior approaches use a two-stage framework that models and controls a system sequentially, whereas Control PINNs incorporate the required optimality conditions in their architecture and loss function. The success of Control PINNs is demonstrated by solving the following open-loop optimal control problems: (i) an analytical problem (ii) a one-dimensional heat equation, and (iii) a two-dimensional predator-prey problem.
- A stochastic covariance shrinkage approach to particle rejuvenation in the ensemble transform particle filterPopov, Andrey A.; Subrahmanya, Amit N.; Sandu, Adrian (Copernicus, 2022-06-22)Rejuvenation in particle filters is necessary to prevent the collapse of the weights when the number of particles is insufficient to properly sample the high-probability regions of the state space. Rejuvenation is often implemented in a heuristic manner by the addition of random noise that widens the support of the ensemble. This work aims at improving canonical rejuvenation methodology by the introduction of additional prior information obtained from climatological samples; the dynamical particles used for importance sampling are augmented with samples obtained from stochastic covariance shrinkage. A localized variant of the proposed method is developed. Numerical experiments with the Lorenz '63 model show that modified filters significantly improve the analyses for low dynamical ensemble sizes. Furthermore, localization experiments with the Lorenz '96 model show that the proposed methodology is extendable to larger systems.
- A Stochastic Covariance Shrinkage Approach to Particle Rejuvenation in the Ensemble Transform Particle FilterPopov, Andrey A.; Subrahmanya, Amit N.; Sandu, Adrian (2021-09-20)Rejuvenation in particle filters is necessary to prevent the collapse of the weights when the number of particles is insufficient to sample the high probability regions of the state space. Rejuvenation is often implemented in a heuristic manner by the addition of stochastic samples that widen the support of the ensemble. This work aims at improving canonical rejuvenation methodology by the introduction of additional prior information obtained from climatological samples; the dynamical particles used for importance sampling are augmented with samples obtained from stochastic covariance shrinkage. The ensemble transport particle filter, and its second order variant, are extended with the proposed rejuvenation approach. Numerical experiments show that modified filters significantly improve the analyses for low dynamical ensemble sizes.