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A variety of complex biological, natural and man-made systems exhibit non-Markovian dynamics that can be modeled through fractional order differential equations, yet, we lack sample comlexity aware system identification strategies. Towards…

Effective data-driven PDE forecasting methods often rely on fixed spatial and / or temporal discretizations. This raises limitations in real-world applications like weather prediction where flexible extrapolation at arbitrary spatiotemporal…

Complex chaotic dynamics, seen in natural and industrial systems like turbulent flows and weather patterns, often span vast spatial domains with interactions across scales. Accurately capturing these features requires a high-dimensional…

In many fields including cosmology, statistical inference often relies on Gaussian likelihoods whose covariance matrices are estimated from a finite number of simulations. This finite-sample estimation introduces noise into the covariance,…

Random dynamical systems (RDS) evolve by a dynamical rule chosen independently with a certain probability, from a given set of deterministic rules. These dynamical systems in an interval reach a steady state with a unique well-defined…

This paper proposes a novel framework for the distributionally robust input and state estimation (DRISE) for autonomous vehicles operating under model uncertainties and measurement outliers. The proposed framework improves the input and…

We demonstrate a data-driven method to solve for the invariant probability density function of a randomly perturbed dynamical system. The key idea is to replace the boundary condition of numerical schemes by a least squares problem…

Data assimilation plays a crucial role in modern weather prediction, providing a systematic way to incorporate observational data into complex dynamical models. The paper addresses continuous data assimilation for a model arising as a…

In many real-world dynamical systems, obtaining precise models of system uncertainty remains a challenge. It may be difficult to estimate noise distributions or robustness bounds, especially when the distributions/robustness bounds vary…

Most signal processing problems involve the challenging task of multidimensional probability density function (PDF) estimation. In this work, we propose a solution to this problem by using a family of Rotation-based Iterative…

In this contribution, we propose a detailed study of interpolation-based data-driven methods that are of relevance in the model reduction and also in the systems and control communities. The data are given by samples of the transfer…

Multiscale dynamics are ubiquitous in applications of modern science. Because of time scale separation between relatively small set of slowly evolving variables and (typically) much larger set of rapidly changing variables, direct numerical…

Physics-informed neural networks are developed to characterize the state of dynamical systems in a random environment. The neural network approximates the probability density function (pdf) or the characteristic function (chf) of the state…

This thesis develops exact analytical tools to study strongly correlated stochastic systems, with a focus on extreme value statistics, gap statistics, and full counting statistics in multi-particle processes. A central contribution is the…

We develop an interpolation-based framework for noisy linear systems with unknown system matrix with bounded norm (implying bounded growth or non-increasing energy), and bounded process noise energy. The proposed approach characterizes all…

Many real-world systems are modelled using complex ordinary differential equations (ODEs). However, the dimensionality of these systems can make them challenging to analyze. Dimensionality reduction techniques like Proper Orthogonal…

Noise fundamentally limits the performance and predictive capabilities of classical and quantum dynamical systems by degrading stability and obscuring intrinsic dynamical characteristics. Characterizing such noise accurately is essential…

This paper compiles several aspects of the dynamics of stochastic approximation algorithms with Markov iterate-dependent noise when the iterates are not known to be stable beforehand. We achieve the same by extending the lock-in probability…

We consider the problem of data-driven stochastic optimal control of an unknown LTI dynamical system. Assuming the process noise is normally distributed, we pose the problem of steering the state's mean and covariance to a target normal…

This paper proposes a comprehensive and unprecedented framework that streamlines the derivation of exact, compact -- yet tractable -- solutions for the probability density function (PDF) and cumulative distribution function (CDF) of the sum…