Nonlocal Partial Differential Equations: entropies, gradient flows, phase transitions and applications
Lead Research Organisation:
Imperial College London
Department Name: Mathematics
Abstract
Understanding the qualitative properties of large systems of interacting particles is of crucial importance in many applications in physics and biology such as molecular dynamic simulations, particles immersed in a fluid, organogenesis modelling, swarming methods for optimization or herding in the social sciences and models for opinion formation, to name a few. This project will be devoted to the further advancement in our understanding of the connection between particle descriptions and continuum models via the passage to the thermodynamic limit. One of our main goals will be to study the thermodynamic (mean field) limit for systems of interacting particles in rugged, multiscale energy landscapes, of the type that one frequently encounters in applications such as biophysics and chemical physics. The dynamics in such a potential exhibit metastable behavior at all scales. In particular, we want to understand the effect of the multiscale structure on the existence and stability of stationary states of the mean field dynamics. Phase transitions, i.e., abrupt changes of behavior, driven by noise will be analyzed for rugged energy landscapes. We will then employ tools from control theory in order to develop algorithms for stabilizing unstable steady states. In addition, we will develop efficient numerical techniques for solving nonlocal, nonlinear mean field equations and we will apply them to diverse problems such as dynamical density functional theory and mathematical models from the social sciences, including models for opinion formation. Furthermore, we will use appropriate systems of interacting particles and their corresponding mean field limit in order to develop consensus-based global optimization algorithms that can be applied to potentials with a multiscale structure characterized by (infinitely) many local minima.
Planned Impact
It is anticipated that the successful completion of the proposed research will have substantial impact to the academic community, in particular to researchers working in several areas of applied mathematics and mathematical physics, including the theory of partial differential equations, statistical mechanics and optimization and control. Furthermore, the computational algorithms that we will develop will be of use to practitioners working in computational physics and in the mathematical modeling in the social sciences.
Methods for Communication.- The team will have a common strategy for disseminating and communicating the research results that we detail next:
Publications.- The PI and Co-PI have an outstanding publication record with articles in the leading international research journals in applied mathematics and mathematical physics. It is anticipated that the work outlined in this proposal with be submitted to top journals in mathematics, applied mathematics, mathematical physics, and mathematical biology.
Workshops and Scientific Events.- An important route to obtain feedback and to engage the closest scientific community in this topic will be a dedicated workshop that we will organise and host to be held provisionally in Spring 2019. This workshop will count with the top-world researchers in the field. We will organize annual small (2-3 days, 20 participants) focused workshops in which we plan to invite researchers mostly from the UK and European countries to increase the number of collaborations at Imperial with top groups in near geographical areas. These small workshops will be funded from visitors expenses already budgeted.
The PI and Co-PI have considerable experience in organising successful workshops and summer schools as detailed in the track of record. Furthermore, 2018 will be the year of mathematical biology in Europe organized by the EMS and the ESMTB in which JAC is one of the main organisers as chairman of the Applied Mathematics Committee of the European Mathematical Society. During that year there will be several events organized across Europe in which this team will be involved trying to understand how swarming models can be used in organogenesis and in other instances in mathematical biology through experimental real data.
Collaborations and Visitors.- Additionally the PI and Co-PI have an active and extensive network of international collaborations with visitors planned to come to Imperial, and the investigators of the team will be visiting regularly research groups in the UK and overseas. This will aid in the dissemination of the research and increase its impact through the academic community. This is ensured by the high visibility of the research done as measured by the number of researchers citing publications of the PI and Co-PI.
Public engagement: communicating research and scientific values.- Imperial College has an efficient press office promoting innovative research conducted at the College. With their help, we plan to promote the numerous aspects of the research that have significant impact and that would be of interest to the public. This includes initiating contact with interested media sources, and possibly traveling to schools around London area to promote mathematics, and the research proposed here in particular. Furthermore, we plan to write popular science articles and to give popular science talks on the applications of applied mathematics and of statistical physics to the social sciences.
Methods for Communication.- The team will have a common strategy for disseminating and communicating the research results that we detail next:
Publications.- The PI and Co-PI have an outstanding publication record with articles in the leading international research journals in applied mathematics and mathematical physics. It is anticipated that the work outlined in this proposal with be submitted to top journals in mathematics, applied mathematics, mathematical physics, and mathematical biology.
Workshops and Scientific Events.- An important route to obtain feedback and to engage the closest scientific community in this topic will be a dedicated workshop that we will organise and host to be held provisionally in Spring 2019. This workshop will count with the top-world researchers in the field. We will organize annual small (2-3 days, 20 participants) focused workshops in which we plan to invite researchers mostly from the UK and European countries to increase the number of collaborations at Imperial with top groups in near geographical areas. These small workshops will be funded from visitors expenses already budgeted.
The PI and Co-PI have considerable experience in organising successful workshops and summer schools as detailed in the track of record. Furthermore, 2018 will be the year of mathematical biology in Europe organized by the EMS and the ESMTB in which JAC is one of the main organisers as chairman of the Applied Mathematics Committee of the European Mathematical Society. During that year there will be several events organized across Europe in which this team will be involved trying to understand how swarming models can be used in organogenesis and in other instances in mathematical biology through experimental real data.
Collaborations and Visitors.- Additionally the PI and Co-PI have an active and extensive network of international collaborations with visitors planned to come to Imperial, and the investigators of the team will be visiting regularly research groups in the UK and overseas. This will aid in the dissemination of the research and increase its impact through the academic community. This is ensured by the high visibility of the research done as measured by the number of researchers citing publications of the PI and Co-PI.
Public engagement: communicating research and scientific values.- Imperial College has an efficient press office promoting innovative research conducted at the College. With their help, we plan to promote the numerous aspects of the research that have significant impact and that would be of interest to the public. This includes initiating contact with interested media sources, and possibly traveling to schools around London area to promote mathematics, and the research proposed here in particular. Furthermore, we plan to write popular science articles and to give popular science talks on the applications of applied mathematics and of statistical physics to the social sciences.
Publications
Abdulle A
(2019)
Accelerated convergence to equilibrium and reduced asymptotic variance for Langevin dynamics using Stratonovich perturbations
in Comptes Rendus. Mathématique
Abdulle A
(2022)
Eigenfunction martingale estimating functions and filtered data for drift estimation of discretely observed multiscale diffusions.
in Statistics and computing
Abdulle A
(2021)
Drift Estimation of Multiscale Diffusions Based on Filtered Data
in Foundations of Computational Mathematics
Aceves-Sánchez P
(2019)
Hydrodynamic limits for kinetic flocking models of Cucker-Smale type.
in Mathematical biosciences and engineering : MBE
Aceves-Sánchez P
(2019)
Hydrodynamic limits for kinetic flocking models of Cucker-Smale type
Alasio L
(2020)
The role of a strong confining potential in a nonlinear Fokker-Planck equation
in Nonlinear Analysis
Antonio Carrillo J
(2019)
A Lipschitz metric for the Hunter-Saxton equation
in Communications in Partial Differential Equations
Description | The rigorous analysis of models for opinion formation and consensus formation, in particular the effect of boundary conditions. The rigorous proof that these models exhibit phase transitions and the complete local and global bifurcation theory. The multiscale analysis of systems of interacting particles in a two-scale confining potential. The development of global optimization algorithms based on consensus formation for systems of interacting agents. The development of spectral numerical methods for the solution of mean field PDEs without assuming an underlying gradient structure. The study of optimal algorithms for sampling and optimization based on ensemble methods. The development of inference methodologies of interacting particle systems. The study of the breakdown of linear response theory for mean field PDEs exhibiting phase transitions. The study of fluctuations around the mean field limit, in particular in the presence of phase transitions. The development of a purely functional analytic characterization of phase transitions for mean field limits of weakly interacting diffusions. |
Exploitation Route | They can lead to efficient algorithms for global optimization. These algorithms can in turn be used in the training of neural networks, which is a very important ingredient of machine learning algorithms. One of the most important outcomes of this funding was the development of a rigorous theory for the study of phase transitions and of fluctuations for mean field PDEs. The theoretical framework that was developed due to this funding has already been used by several researchers in the areas of partial differential equations and statistical mechanics. They can lead to the development of efficient methodologies for sampling. The algorithms that were developed due to this funding have already been used to develop efficient Bayesian methodologies for solving inverse problems for PDEs. They can be used to understand phenomena in the social sciences, such as opinion formation. They can be used in order to develop efficient optimal control algorithms for mean field models. |
Sectors | Financial Services and Management Consultancy Healthcare Government Democracy and Justice Transport |
URL | http://wwwf.imperial.ac.uk/~pavl/ |
Description | The findings of this research project have led to further research related to algorithms for global optimization, sampling from probability distributions in high dimensional spaces, the development of inference methodologies for dynamics on networks, the analysis of models for systemic risk and the stability of large financial networks. The continuation of this EPSRC-funded projects in now funded by JP Morgan. We are in close contact with the Artificial Intelligence (AI) group at GP Morgan, and one of the PhD students of our group will do an internship there. We are aiming at developing algorithms for the efficient training of neural networks, in connection to algorithmic training. We are also discussing with other investment banks such as Standard Chartered. I expect that the outcomes of this award will have considerable impact in areas such as artificial intelligence and machine learning, the modeling of financial networks and mathematical modeling in the social sciences. |
First Year Of Impact | 2019 |
Sector | Financial Services, and Management Consultancy,Healthcare,Government, Democracy and Justice |
Impact Types | Economic Policy & public services |
Description | Mathematics of Machine Learning and Global Optimization |
Organisation | Imperial College London |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | Joint project with Dr P. Panos (ICL, Computer Science) and Dr N. Kantas (ICL, Mathematics, Statistics section) on the analysis of gradient descent algorithms used in optimization and machine learning. The main goal is to explore the stability and robustness of such algorithms using tools from the the optimal control theory for nonlinear PDEs. This project has already led to a successful research proposal "Dynamics, Control and Uncertainty Quantification for Stable Machine Learning Algorithms" to be funded by J.P. Morgan that provides us with funds for hiring a research associate for 12 months. |
Collaborator Contribution | Joint writing of the successful proposal "Dynamics, Control and Uncertainty Quantification for Stable Machine Learning Algorithms". |
Impact | Funding from J.P. Morgan co-supervision of a PhD student |
Start Year | 2018 |