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Linear dynamical systems are canonical models for learning-based control of plants with uncertain dynamics. The setting consists of a stochastic differential equation that captures the state evolution of the plant understudy, while the true…
A long-standing problem at the interface of artificial intelligence and applied mathematics is to devise an algorithm capable of achieving human level or even superhuman proficiency in transforming observed data into predictive mathematical…
One's ability to learn a generative model of the world without supervision depends on the extent to which one can construct abstract knowledge representations that generalize across experiences. To this end, capturing an accurate…
Modeling living systems at the collective scale can be very challenging because the individual constituents can themselves be complex and the respective interactions between the constituents are not fully understood. With the advent of high…
Understanding real-world dynamical phenomena remains a challenging task. Across various scientific disciplines, machine learning has advanced as the go-to technology to analyze nonlinear dynamical systems, identify patterns in big data, and…
Partial differential equations (PDEs) that fit scientific data can represent physical laws with explainable mechanisms for various mathematically-oriented subjects, such as physics and finance. The data-driven discovery of PDEs from…
While existing mathematical descriptions can accurately account for phenomena at microscopic scales (e.g. molecular dynamics), these are often high-dimensional, stochastic and their applicability over macroscopic time scales of physical…
The solutions of Hamiltonian equations are known to describe the underlying phase space of a mechanical system. In this article, we propose a novel spatio-temporal model using a strategic modification of the Hamiltonian equations,…
This paper introduces a linear state-space model with time-varying dynamics. The time dependency is obtained by forming the state dynamics matrix as a time-varying linear combination of a set of matrices. The time dependency of the weights…
Inverse problems with spatiotemporal observations are ubiquitous in scientific studies and engineering applications. In these spatiotemporal inverse problems, observed multivariate time series are used to infer parameters of physical or…
Stochastic process models for spatiotemporal data underlying random fields find substantial utility in a range of scientific disciplines. Subsequent to predictive inference on the values of the random field (or spatial surface indexed…
Change points in real-world systems mark significant regime shifts in system dynamics, possibly triggered by exogenous or endogenous factors. These points define regimes for the time evolution of the system and are crucial for understanding…
We study the problem of identifying the dynamics of a linear system when one has access to samples generated by a similar (but not identical) system, in addition to data from the true system. We use a weighted least squares approach and…
Throughout the history of science, physics-based modeling has relied on judiciously approximating observed dynamics as a balance between a few dominant processes. However, this traditional approach is mathematically cumbersome and only…
Modeling complex spatiotemporal dynamics, particularly in far-from-equilibrium systems, remains a grand challenge in science. The governing partial differential equations (PDEs) for these systems are often intractable to derive from first…
Differential equations and numerical methods are extensively used to model various real-world phenomena in science and engineering. With modern developments, we aim to find the underlying differential equation from a single observation of…
Understanding how systems evolve over time often requires discovering the differential equations that govern their behavior. Automatically learning these equations from experimental data is challenging when the data are noisy or limited,…
With the advent of modern data collection and storage technologies, data-driven approaches have been developed for discovering the governing partial differential equations (PDE) of physical problems. However, in the extant works the model…
Physics-based and first-principles models pervade the engineering and physical sciences, allowing for the ability to model the dynamics of complex systems with a prescribed accuracy. The approximations used in deriving governing equations…
Dynamical systems theory describes how interacting quantities change over time and space, from molecular oscillators to large-scale biological patterns. Such systems often involve nonlinear feedbacks, delays, and interactions across scales.…