arXiv subjects

Andrey A. Popov

Publications and source records attributed to Andrey A. Popov.

At least 19 recordsLinked to original sources

arXiv2026

Unbiased estimation of squared concentration in the Fisher-von Mises-Langevin distribution and the impossibility of unbiased concentration

The estimation of concentration parameter in Fisher-von Mises-Langevin distribution is the directional statistics analogue of the estimation of the precision matrix for the Gaussian distribution. In this work we show that unbiased estimation of this parameter is impossible. With this realization in hand, we provide an alternative parameterization of the Fisher-von Mises-Langevin distribution in terms of the squared concentration, which we term the intensity. We fruther show that unbiased estimation of thereof is possible, and provide (almost) unbiased estimators thereof in terms of a partial sum U-statistic. We showcase our new estimator on synthetic data, New York taxi trip data, and on spherical word embeddings.

math.ST↗

arXiv2026

Pose Tracking with a Foundation Pose Model and an Ensemble Directional Kalman Filter

This paper introduces the ensemble directional Kalman filter (EnDKF), an ensemble-based Kalman filtering approach for pose tracking that jointly estimates an object's position and attitude using ideas from directional statistics. The EnDKF integrates a unit-quaternion attitude representation to move beyond canonical Kalman filter mean and covariance assumptions that poorly capture directional uncertainty. Experiments on a synthetic constant-velocity constant-angular-velocity system and a digital-twin head-tracking scenario using the FoundationPose algorithm demonstrate a significant reduction in error as opposed to merely using measurements.

cs.LG↗

arXiv2026

Learning Discriminators for Resampling in the Ensemble Gaussian Mixture Filter through a Normalizing Flow Approach

The ensemble Gaussian mixture filter (EnGMF) is a powerful, convergent particle filter capable of medium-to-high dimensional non-linear filtering. The EnGMF relies on a resampling step that can generate physically unrealistic posterior samples, that would subsequently produce physically meaningless forecasts. This work introduces the discriminator-informed resampling procedure, that augments the posterior resampling step with a discriminator that accepts or rejects candidate particles based on their physical plausibility. In this work these discriminators are learned through a normalizing flow approach. Numerical experiments on both the Ikeda map and the Lorenz '63 system show that discriminator informed resampling procedure consistently reduces error relative to the standard EnGMF in low-ensemble regimes.

cs.LG↗

arXiv2026

Learning to Trust AI and Data-driven models in Data Assimilation through a Multifidelity Ensemble Gaussian Mixture Filter Framework

AI and data-driven models have large potential for data assimilation applications by creating fast and accurate forecasts. Their tendency to produce spurious inaccurate, nonphysical results -- hallucination -- however, raises a serious question about their long-term use, and can be categorized as untrustworthy methods. Theory-driven methods on the other hand are slow, but are capable of staying physically realistic due to their mathematical underpinning, and can be categorized as trustworthy methods. We argue that by making use of these methods in tandem, it is possible to build a relative measure of trust between the theory-driven and data-driven methods that results in a combined trustworthy methodology. We argue, and then show, that the bandwidth scaling factors in the kernel density estimates can be used to represent our trust in the theory-driven and data-driven models. We provide for ways in which these measures of trust can be adaptively computed through an expectation-maximization approach. We combine all of these ideas to create the multifidelity ensemble Gaussian mixture filter and its adaptive trust version, which are particle filters capable of high-dimensional data assimilation. We validate our ideas on both a static banana problem and on a sequential filtering example with the Lorenz '96 equations, showing that it is possible to create a particle filter that is capable of high dimensional convergent inference in the undersampled regime -- when the number of theory-driven samples is less than the dimension of the system.

cs.CE↗

arXiv2026

A divide and conquer strategy for multinomial particle filter resampling

This work provides a new multinomial resampling procedure for particle filter resampling, focused on the case where the number of samples required is less than or equal to the size of the underlying discrete distribution. This setting is common in ensemble mixture model filters such as the Gaussian mixture filter. We show superiority of our approach with respect two of the best known multinomial sampling procedures both through a computational complexity analysis and through a numerical experiment.

cs.DS↗

arXiv2025

Ensemble based Closed-Loop Optimal Control using Physics-Informed Neural Networks

The objective of designing a control system is to steer a dynamical system with a control signal, guiding it to exhibit the desired behavior. The Hamilton-Jacobi-Bellman (HJB) partial differential equation offers a framework for optimal control system design. However, numerical solutions to this equation are computationally intensive, and analytical solutions are frequently unavailable. Knowledge-guided machine learning methodologies, such as physics-informed neural networks (PINNs), offer new alternative approaches that can alleviate the difficulties of solving the HJB equation numerically. This work presents a multistage ensemble framework to learn the optimal cost-to-go, and subsequently the corresponding optimal control signal, through the HJB equation. Prior PINN-based approaches rely on a stabilizing the HJB enforcement during training. Our framework does not use stabilizer terms and offers a means of controlling the nonlinear system, via either a singular learned control signal or an ensemble control signal policy. Success is demonstrated in closed-loop control, using both ensemble- and singular-control, of a steady-state time-invariant two-state continuous nonlinear system with an infinite time horizon, accounting of noisy, perturbed system states and varying initial conditions.

cs.LG↗

arXiv2025

Downscaling human mobility data based on demographic socioeconomic and commuting characteristics using interpretable machine learning methods

Understanding urban human mobility patterns at various spatial levels is essential for social science. This study presents a machine learning framework to downscale origin-destination (OD) taxi trips flows in New York City from a larger spatial unit to a smaller spatial unit. First, correlations between OD trips and demographic, socioeconomic, and commuting characteristics are developed using four models: Linear Regression (LR), Random Forest (RF), Support Vector Machine (SVM), and Neural Networks (NN). Second, a perturbation-based sensitivity analysis is applied to interpret variable importance for nonlinear models. The results show that the linear regression model failed to capture the complex variable interactions. While NN performs best with the training and testing datasets, SVM shows the best generalization ability in downscaling performance. The methodology presented in this study provides both analytical advancement and practical applications to improve transportation services and urban development.

cs.LG↗

arXiv2025

A copula-based rank histogram ensemble filter

Serial ensemble filters implement triangular probability transport maps to reduce high-dimensional inference problems to sequences of state-by-state univariate inference problems. The univariate inference problems are solved by sampling posterior probability densities obtained by combining constructed prior densities with observational likelihoods according to Bayes' rule. Many serial filters in the literature focus on representing the marginal posterior densities of each state. However, rigorously capturing the conditional dependencies between the different univariate inferences is crucial to correctly sampling multidimensional posteriors. This work proposes a new serial ensemble filter, called the copula rank histogram filter (CoRHF), that seeks to capture the conditional dependency structure between variables via empirical copula estimates; these estimates are used to rigorously implement the triangular (state-by-state univariate) Bayesian inference. The success of the CoRHF is demonstrated on two-dimensional examples and the Lorenz '63 problem. A practical extension to the high-dimensional setting is developed by localizing the empirical copula estimation, and is demonstrated on the Lorenz '96 problem.

stat.ME↗

arXiv2024

Improving the Adaptive Moment Estimation (ADAM) stochastic optimizer through an Implicit-Explicit (IMEX) time-stepping approach

The Adam optimizer, often used in Machine Learning for neural network training, corresponds to an underlying ordinary differential equation (ODE) in the limit of very small learning rates. This work shows that the classical Adam algorithm is a first-order implicit-explicit (IMEX) Euler discretization of the underlying ODE. Employing the time discretization point of view, we propose new extensions of the Adam scheme obtained by using higher-order IMEX methods to solve the ODE. Based on this approach, we derive a new optimization algorithm for neural network training that performs better than classical Adam on several regression and classification problems.

cs.CE↗

arXiv2024

The Ensemble Epanechnikov Mixture Filter

In the high-dimensional setting, Gaussian mixture kernel density estimates become increasingly suboptimal. In this work we aim to show that it is practical to instead use the optimal multivariate Epanechnikov kernel. We make use of this optimal Epanechnikov mixture kernel density estimate for the sequential filtering scenario through what we term the ensemble Epanechnikov mixture filter (EnEMF). We provide a practical implementation of the EnEMF that is as cost efficient as the comparable ensemble Gaussian mixture filter. We show on a static example that the EnEMF is robust to growth in dimension, and also that the EnEMF has a significant reduction in error per particle on the 40-variable Lorenz '96 system.

stat.ML↗

arXiv2024

Ensemble-localized Kernel Density Estimation with Applications to the Ensemble Gaussian Mixture Filter

The ensemble Gaussian mixture filter (EnGMF) is a non-linear filter suited to data assimilation of highly non-Gaussian and non-linear models that has practical utility in the case of a small number of samples, and theoretical convergence to full Bayesian inference in the ensemble limit. We aim to increase the utility of the EnGMF by introducing an ensemble-local notion of covariance into the kernel density estimation (KDE) step for the prior distribution. We prove that in the Gaussian case, our new ensemble-localized KDE technique is exactly the same as more traditional KDE techniques. We also show an example of a non-Gaussian distribution that can fail to be approximated by canonical KDE methods, but can be approximated well by our new KDE technique. We showcase our new KDE technique on a simple bivariate problem, showing that it has nice qualitative and quantitative properties, and significantly improves the estimate of the prior and posterior distributions for all ensemble sizes tested. We additionally show the utility of the proposed methodology for sequential filtering for the Lorenz '63 equations, achieving a significant reduction in error, and less conservative behavior in the uncertainty estimate with respect to traditional techniques.

math.OC↗

arXiv2024

Precision Mars Entry Navigation with Atmospheric Density Adaptation via Neural Networks

Spacecraft entering Mars require precise navigation algorithms capable of accurately estimating the vehicle's position and velocity in dynamic and uncertain atmospheric environments. Discrepancies between the true Martian atmospheric density and the onboard density model can significantly impair the performance of spacecraft entry navigation filters. This work introduces a new approach to online filtering for Martian entry using a neural network to estimate atmospheric density and employing a consider analysis to account for the uncertainty in the estimate. The network is trained on an exponential atmospheric density model, and its parameters are dynamically adapted in real time to account for any mismatch between the true and estimated densities. The adaptation of the network is formulated as a maximum likelihood problem by leveraging the measurement innovations of the filter to identify optimal network parameters. Within the context of the maximum likelihood approach, incorporating a neural network enables the use of stochastic optimizers known for their efficiency in the machine learning domain. Performance comparisons are conducted against two online adaptive approaches, covariance matching and state augmentation and correction, in various realistic Martian entry navigation scenarios. The results show superior estimation accuracy compared to other approaches, and precise alignment of the estimated density with a broad selection of realistic Martian atmospheres sampled from perturbed Mars-GRAM data.

cs.LG↗

arXiv2024

What are You Weighting For? Improved Weights for Gaussian Mixture Filtering With Application to Cislunar Orbit Determination

This work focuses on the critical aspect of accurate weight computation during the measurement incorporation phase of Gaussian mixture filters. The proposed novel approach computes weights by linearizing the measurement model about each component's posterior estimate rather than the the prior, as traditionally done. This work proves equivalence with traditional methods for linear models, provides novel sigma-point extensions to the traditional and proposed methods, and empirically demonstrates improved performance in nonlinear cases. Two illustrative examples, the Avocado and a cislunar single target tracking scenario, serve to highlight the advantages of the new weight computation technique by analyzing filter accuracy and consistency through varying the number of Gaussian mixture components.

stat.ME↗

arXiv2024

Preserving Nonlinear Constraints in Variational Flow Filtering Data Assimilation

Data assimilation aims to estimate the states of a dynamical system by optimally combining sparse and noisy observations of the physical system with uncertain forecasts produced by a computational model. The states of many dynamical systems of interest obey nonlinear physical constraints, and the corresponding dynamics is confined to a certain sub-manifold of the state space. Standard data assimilation techniques applied to such systems yield posterior states lying outside the manifold, violating the physical constraints. This work focuses on particle flow filters which use stochastic differential equations to evolve state samples from a prior distribution to samples from an observation-informed posterior distribution. The variational Fokker-Planck (VFP) -- a generic particle flow filtering framework -- is extended to incorporate non-linear, equality state constraints in the analysis. To this end, two algorithmic approaches that modify the VFP stochastic differential equation are discussed: (i) VFPSTAB, to inexactly preserve constraints with the addition of a stabilizing drift term, and (ii) VFPDAE, to exactly preserve constraints by treating the VFP dynamics as a stochastic differential-algebraic equation (SDAE). Additionally, an implicit-explicit time integrator is developed to evolve the VFPDAE dynamics. The strength of the proposed approach for constraint preservation in data assimilation is demonstrated on three test problems: the double pendulum, Korteweg-de-Vries, and the incompressible Navier-Stokes equations.

math.OC↗

arXiv2024

A Sensor-Based Simulation Method for Spatiotemporal Event Detection

Human movements in urban areas are essential to understand human-environment interactions. However, activities and associated movements are full of uncertainties due to the complexity of a city. In this paper, we propose a novel sensor-based approach for spatiotemporal event detection based on the Discrete Empirical Interpolation Method. Specifically, we first identify the key locations, defined as 'sensors' , which have the strongest correlation with the whole dataset. We then simulate a regular uneventful scenario with the observation data points from those key lo-cations. By comparing the simulated and observation scenarios, events are extracted both spatially and temporally. We apply this method in New York City with taxi trip record data. Results show that this method is effective in detecting when and where events occur.

cs.IR↗

arXiv2023

Bayesian Recursive Update for Ensemble Kalman Filters

Few real-world systems are amenable to truly Bayesian filtering; nonlinearities and non-Gaussian noises can wreak havoc on filters that rely on linearization and Gaussian uncertainty approximations. This article presents the Bayesian Recursive Update Filter (BRUF), a Kalman filter that uses a recursive approach to incorporate information from nonlinear measurements. The BRUF relaxes the measurement linearity assumption of the Extended Kalman Filter (EKF) by dividing the measurement update into a user-defined number of steps. The proposed technique is extended for ensemble filters in the Bayesian Recursive Update Ensemble Kalman Filter (BRUEnKF). The performance of both filters is demonstrated in numerical examples, and new filters are introduced which exploit the theoretical foundation of the BRUF in different ways. A comparison between the BRUEnKF and Gromov flow, a popular particle flow algorithm, is presented in detail. Finally, the BRUEnKF is shown to outperform the EnKF for a very high-dimensional system.

math.NA↗

arXiv2023

Small-data Reduced Order Modeling of Chaotic Dynamics through SyCo-AE: Synthetically Constrained Autoencoders

Data-driven reduced order modeling of chaotic dynamics can result in systems that either dissipate or diverge catastrophically. Leveraging non-linear dimensionality reduction of autoencoders and the freedom of non-linear operator inference with neural-networks, we aim to solve this problem by imposing a synthetic constraint in the reduced order space. The synthetic constraint allows our reduced order model both the freedom to remain fully non-linear and highly unstable while preventing divergence. We illustrate the methodology with the classical 40-variable Lorenz '96 equations, showing that our methodology is capable of producing medium-to-long range forecasts with lower error using less data.

cs.LG↗

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↗