arXiv subjects

Andrey A. Popov

Publications and source records attributed to Andrey A. Popov.

23 records · Page 2Linked to original sources

arXiv2022

Adjoint-Matching Neural Network Surrogates for Fast 4D-Var Data Assimilation

Data assimilation is the process of fusing information from imperfect computer simulations with noisy, sparse measurements of reality to obtain improved estimates of the state or parameters of a dynamical system of interest. The data assimilation procedures used in many geoscience applications, such as numerical weather forecasting, are variants of the our-dimensional variational (4D-Var) algorithm. The cost of solving the underlying 4D-Var optimization problem is dominated by the cost of repeated forward and adjoint model runs. This motivates substituting the evaluations of the physical model and its adjoint 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 solution depends on the accuracy with each the surrogate captures both the forward and the adjoint model dynamics. We formulate and analyze several approaches to incorporate adjoint information into the construction of neural network surrogates. The resulting networks are tested on unseen data and in a sequential data assimilation problem using the Lorenz-63 system. Surrogates constructed using adjoint information demonstrate superior performance on the 4D-Var data assimilation problem compared to a standard neural network surrogate that uses only forward dynamics information.

cs.LG↗

arXiv2022

Physics-informed neural networks for PDE-constrained optimization and control

A fundamental problem in science and engineering is designing optimal control policies that steer a given system towards a desired outcome. This work proposes Control Physics-Informed Neural Networks (Control PINNs) that simultaneously solve for a given system state, and for the optimal control signal, in a one-stage framework that conforms to the underlying physical laws. Prior approaches use a two-stage framework that first models and then controls a system in sequential order. In contrast, a Control PINN incorporates the required optimality conditions in its architecture and in its 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.

cs.LG↗

arXiv2022

A Meta-learning Formulation of the Autoencoder Problem for Non-linear Dimensionality Reduction

A rapidly growing area of research is the use of machine learning approaches such as autoencoders for dimensionality reduction of data and models in scientific applications. We show that the canonical formulation of autoencoders suffers from several deficiencies that can hinder their performance. Using a meta-learning approach, we reformulate the autoencoder problem as a bi-level optimization procedure that explicitly solves the dimensionality reduction task. We prove that the new formulation corrects the identified deficiencies with canonical autoencoders, provide a practical way to solve it, and showcase the strength of this formulation with a simple numerical illustration.

cs.LG↗

arXiv2021

Investigation of Nonlinear Model Order Reduction of the Quasigeostrophic Equations through a Physics-Informed Convolutional Autoencoder

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.

cs.LG↗

arXiv2019

ODE Test Problems: a MATLAB suite of initial value problems

ODE Test Problems (OTP) is an object-oriented MATLAB package offering a broad range of initial value problems which can be used to test numerical methods such as time integration methods and data assimilation (DA) methods. It includes problems that are linear and nonlinear, homogeneous and nonhomogeneous, autonomous and nonautonomous, scalar and high-dimensional, stiff and nonstiff, and chaotic and nonchaotic. Many are real-world problems from fields such as chemistry, astrophysics, meteorology, and electrical engineering. OTP also supports partitioned ODEs for testing IMEX methods, multirate methods, and other multimethods. Functions for plotting solutions and creating movies are available for all problems, and exact solutions are provided when available. OTP is desgined for ease of use-meaning that working with and modifying problems is simple and intuitive.

math.NA↗