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A growing body of literature has been leveraging techniques of machine learning (ML) to build novel approaches to approximating the solutions to partial differential equations. Noticeably absent from the literature is a systematic…

The spatial organization of individuals and their interactions in communities are important factors known to preserve diversity in many complex systems. Inspired by metapopulation models from ecology, we study opinion formation using a…

Diffusion models, a powerful and universal generative AI technology, have achieved tremendous success in computer vision, audio, reinforcement learning, and computational biology. In these applications, diffusion models provide flexible…

We consider distributed multitask learning problems over a network of agents where each agent is interested in estimating its own parameter vector, also called task, and where the tasks at neighboring agents are related according to a set…

Building Virtual Cells that can accurately simulate cellular responses to perturbations is a long-standing goal in systems biology. A fundamental challenge is that high-throughput single-cell sequencing is destructive: the same cell cannot…

One of the main challenges in diffusion-based molecular communication is dealing with the non-linearity of reaction-diffusion chemical equations. While numerical methods can be used to solve these equations, a change in the input signals or…

We analyze the dynamics of multi-agent collective behavior models and their control theoretical properties. We first derive a large population limit to parabolic diffusive equations. We also show that the non-local transport equations…

A new simple model of diffusion of innovations in a social network with upgrading costs is introduced. Agents are characterized by a single real variable, their technological level. According to local information agents decide whether to…

Adaptive networks consist of a collection of nodes with adaptation and learning abilities. The nodes interact with each other on a local level and diffuse information across the network to solve estimation and inference tasks in a…

The importance of the ability of predict trends in social media has been growing rapidly in the past few years with the growing dominance of social media in our everyday's life. Whereas many works focus on the detection of anomalies in…

Diffusion models have demonstrated remarkable capabilities in synthesizing realistic images, spurring interest in using their representations for various downstream tasks. To better understand the robustness of these representations, we…

Innovation diffusion in the networked population is an essential process that drives the progress of human society. Despite the recent advances in network science, a fundamental understanding of network properties that regulate such…

We introduce diffusively coupled networks where the dynamical system at each vertex is planar Hamiltonian. The problems we address are synchronisation and an analogue of diffusion-driven Turing instability for time-dependent homogeneous…

Denoising Diffusion models are gaining increasing popularity in the field of generative modeling for several reasons, including the simple and stable training, the excellent generative quality, and the solid probabilistic foundation. In…

This article presents a rigorous mathematical analysis of the Friedkin--Johnsen model of social influence on networks. We frame the opinion dynamics as a discrete boundary-value problem on a network, emphasizing the role of stubborn…

We study how the Einstein relation between spontaneous fluctuations and the response to an external perturbation holds in the absence of currents, for the comb model and the elastic single-file, which are examples of systems with…

Diffusion is a fundamental physical phenomenon with critical applications in fields such as metallurgy, cell biology, and population dynamics. While standard diffusion is well-understood, anomalous diffusion often requires complex non-local…

This paper addresses the problem of generating dynamically admissible trajectories for control tasks using diffusion models, particularly in scenarios where the environment is complex and system dynamics are crucial for practical…

Recent advances in diffusion models enable many powerful instruments for image editing. One of these instruments is text-driven image manipulations: editing semantic attributes of an image according to the provided text description. %…

We study diffusion in a network which is governed by non-autonomous Kirchhoff conditions at the vertices of the graph. Also the diffusion coefficients may depend on time. We prove at first a result on existence and uniqueness using form…