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Identifying accurate dynamic models is required for the simulation and control of various technical systems. In many important real-world applications, however, the two main modeling approaches often fail to meet requirements: first…

Identifying dynamical systems from experimental data is a notably difficult task. Prior knowledge generally helps, but the extent of this knowledge varies with the application, and customized models are often needed. Neural ordinary…

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…

Collective behavior in animals has long been modeled through self-propelled particle models, which reproduce striking group-level phenomena through abstract interaction forces. Yet these models are fundamentally descriptive: they leave open…

The brain-body-environment framework studies adaptive behavior through embodied and situated agents, emphasizing interactions between brains, biomechanics, and environmental dynamics. However, many models often treat the brain as a network…

A variety of computational models have been developed to describe active matter at different length and time scales. The diversity of the methods and the challenges in modeling active matter---ranging from molecular motors and cytoskeletal…

The modern machine learning methods allow one to obtain the data-driven models in various ways. However, the more complex the model is, the harder it is to interpret. In the paper, we describe the algorithm for the mathematical equations…

Humans and animals excel in combining information from multiple sensory modalities, controlling their complex bodies, adapting to growth, failures, or using tools. These capabilities are also highly desirable in robots. They are displayed…

In this paper, we consider the problem of learning prediction models for spatiotemporal physical processes driven by unknown partial differential equations (PDEs). We propose a deep learning framework that learns the underlying dynamics and…

From tumour invasion to cell sorting and animal territoriality, many biological systems rely on nonlocal interactions that drive complex spatial organisation. Partial differential equations (PDEs) with nonlocal advection are increasingly…

Partial differential equations (PDEs) govern nearly every physical process in science and engineering, yet solving them at scale remains prohibitively expensive. Generative AI has transformed language, vision, and protein science, but…

Abridged abstract: Inert interactions between randomly moving entities and spatial disorder play a crucial role in quantifying the diffusive properties of a system. These interactions affect only the movement of the entities, and examples…

Basic principles of mathematical modeling are reviewed in this book, with the focus on physics and its practical applications, and examples of selected mathematical methods are presented. Most of the models have been imported from physics…

Intracellular transport processes are essential to the healthy development of many organisms as well as more generally to healthy cellular function. The complex dynamics and interactions between protein molecules and filaments on different…

With the influx of complex and detailed tracking data gathered from electronic tracking devices, the analysis of animal movement data has recently emerged as a cottage industry amongst biostatisticians. New approaches of ever greater…

Mechanistic modelling of animal movement is often formulated in discrete time despite problems with scale invariance, such as handling irregularly timed observations. A natural solution is to formulate in continuous time, yet uptake of this…

Human motion prediction is an important and challenging topic that has promising prospects in efficient and safe human-robot-interaction systems. Currently, the majority of the human motion prediction algorithms are based on deterministic…

Animals often exhibit changes in their behavior during migration. Telemetry data provide a way to observe geographic position of animals over time, but not necessarily changes in the dynamics of the movement process. Continuous-time models…

Nonlocality is important in realistic mathematical models of physical and biological systems when local models fail to capture the essential dynamics and interactions that occur over a range of distances. This review illustrates different…

Nonlocal aggregation-diffusion models, when coupled with a spatial map, can capture cognitive and memory-based influences on animal movement and population-level patterns. In this work, we study a one-dimensional…