Jose A. Carrillo

Jose A. Carrillo, Bokai Yan

In this work we numerically study the diffusive limit of run & tumble kinetic models for cell motion due to chemotaxis by means of asymptotic preserving schemes. It is well-known that the diffusive limit of these models leads to the classical Patlak-Keller-Segel macroscopic model for chemotaxis. We will show that the proposed scheme is able to accurately approximate the solutions before blow-up time for small parameter. Moreover, the numerical results indicate that the global solutions of the kinetic models stabilize for long times to steady states for all the analyzed parameter range. We also generalize these asymptotic preserving schemes to two dimensional kinetic models in the radial case. The blow-up of solutions is numerically investigated in all these cases.

Torkel E. Loman, Yurij Salmaniw, Antonio Leon Villares et al.

Partial differential equations often contain unknown functions that are difficult or impossible to measure directly, hampering our ability to derive predictions from the model. Workflows for recovering scalar PDE parameters from data are well studied: here we show how similar workflows can be used to recover functions from data. Specifically, we embed neural networks into the PDE and show how, as they are trained on data, they can approximate unknown functions with arbitrary accuracy. Using nonlocal aggregation-diffusion equations as a case study, we recover interaction kernels and external potentials from steady state data. Specifically, we investigate how a wide range of factors, such as the number of available solutions, their properties, sampling density, and measurement noise, affect our ability to successfully recover functions. Our approach is advantageous because it can utilise standard parameter-fitting workflows, and in that the trained PDE can be treated as a normal PDE for purposes such as generating system predictions.

Jose A. Carrillo, Nicolas Garcia Trillos, Sixu Li et al.

Federated learning is an important framework in modern machine learning that seeks to integrate the training of learning models from multiple users, each user having their own local data set, in a way that is sensitive to data privacy and to communication loss constraints. In clustered federated learning, one assumes an additional unknown group structure among users, and the goal is to train models that are useful for each group, rather than simply training a single global model for all users. In this paper, we propose a novel solution to the problem of clustered federated learning that is inspired by ideas in consensus-based optimization (CBO). Our new CBO-type method is based on a system of interacting particles that is oblivious to group memberships. Our model is motivated by rigorous mathematical reasoning, including a mean field analysis describing the large number of particles limit of our particle system, as well as convergence guarantees for the simultaneous global optimization of general non-convex objective functions (corresponding to the loss functions of each cluster of users) in the mean-field regime. Experimental results demonstrate the efficacy of our FedCBO algorithm compared to other state-of-the-art methods and help validate our methodological and theoretical work.

Jose A. Carrillo, Katy Craig, Yao Yao

Given a large ensemble of interacting particles, driven by nonlocal interactions and localized repulsion, the mean-field limit leads to a class of nonlocal, nonlinear partial differential equations known as aggregation-diffusion equations. Over the past fifteen years, aggregation-diffusion equations have become widespread in biological applications and have also attracted significant mathematical interest, due to their competing forces at different length scales. These competing forces lead to rich dynamics, including symmetrization, stabilization, and metastability, as well as sharp dichotomies separating well-posedness from finite time blowup. In the present work, we review known analytical results for aggregation-diffusion equations and consider singular limits of these equations, including the slow diffusion limit, which leads to the constrained aggregation equation, as well as localized aggregation and vanishing diffusion limits, which lead to metastability behavior. We also review the range of numerical methods available for simulating solutions, with special attention devoted to recent advances in deterministic particle methods. We close by applying such a method -- the blob method for diffusion -- to showcase key properties of the dynamics of aggregation-diffusion equations and related singular limits.