Arshed Nabeel

Arshed Nabeel, Ashwin Karichannavar, Shuaib Palathingal et al.

Theoretical studies have shown that stochasticity can affect the dynamics of ecosystems in counter-intuitive ways. However, without knowing the equations governing the dynamics of populations or ecosystems, it is difficult to ascertain the role of stochasticity in real datasets. Therefore, the inverse problem of inferring the governing stochastic equations from datasets is important. Here, we present an equation discovery methodology that takes time series data of state variables as input and outputs a stochastic differential equation. We achieve this by combining traditional approaches from stochastic calculus with the equation-discovery techniques. We demonstrate the generality of the method via several applications. First, we deliberately choose various stochastic models with fundamentally different governing equations; yet they produce nearly identical steady-state distributions. We show that we can recover the correct underlying equations, and thus infer the structure of their stability, accurately from the analysis of time series data alone. We demonstrate our method on two real-world datasets -- fish schooling and single-cell migration -- which have vastly different spatiotemporal scales and dynamics. We illustrate various limitations and potential pitfalls of the method and how to overcome them via diagnostic measures. Finally, we provide our open-source codes via a package named PyDaDDy (Python library for Data Driven Dynamics).

Utkarsh Pratiush, Arshed Nabeel, Vishwesha Guttal et al.

Collective motion is an ubiquitous phenomenon in nature, inspiring engineers, physicists and mathematicians to develop mathematical models and bio-inspired designs. Collective motion at small to medium group sizes ($\sim$10-1000 individuals, also called the `mesoscale'), can show nontrivial features due to stochasticity. Therefore, characterizing both the deterministic and stochastic aspects of the dynamics is crucial in the study of mesoscale collective phenomena. Here, we use a physics-inspired, neural-network based approach to characterize the stochastic group dynamics of interacting individuals, through a stochastic differential equation (SDE) that governs the collective dynamics of the group. We apply this technique on both synthetic and real-world datasets, and identify the deterministic and stochastic aspects of the dynamics using drift and diffusion fields, enabling us to make novel inferences about the nature of order in these systems.

Arshed Nabeel, Danny Raj M

Most real world collectives, including active particles, living cells, and grains, are heterogeneous, where individuals with differing properties interact. The differences among individuals in their intrinsic properties have emergent effects at the group level. It is often of interest to infer how the intrinsic properties differ among the individuals, based on their observed movement patterns. However, the true individual properties may be masked by emergent effects in the collective. We investigate the inference problem in the context of a bidisperse collective with two types of agents, where the goal is to observe the motion of the collective and classify the agents according to their types. Since collective effects such as jamming and clustering affect individual motion, an agent's own movement does not have sufficient information to perform the classification well: a simple observer algorithm, based only on individual velocities cannot accurately estimate the level of heterogeneity of the system, and often misclassifies agents. We propose a novel approach to the classification problem, where collective effects on an agent's motion is explicitly accounted for. We use insights about the physics of collective motion to quantify the effect of the neighbourhood on an agent using a neighbourhood parameter. Such an approach can distinguish between agents of two types, even when their observed motion is identical. This approach estimates the level of heterogeneity much more accurately, and achieves significant improvements in classification. Our results demonstrate that explicitly accounting for neighbourhood effects is often necessary to correctly infer intrinsic properties of individuals.