Gianluca Fabiani


PhD in: MERC
Ciclo: XXXVI

Project title: Machine learning-based Modelling and Numerical Analysis of Emergent Dynamics of Phase Field Models

Supervisor(s): Constantinos Siettos, Yannis Kevrekidis

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  • Gianluca Fabiani is a PhD student in Modeling and Engineering Risk and Complexity (MERC) at Scuola Superiore Meridionale (SSM).

    Gialuca Fabiani holds a Bachelor’s Degree in Mathematics and a Master’s Degree in Applied Mathematics from the department of mathematics and applications “Renato Caccioppoli”, University of Naples Federico II.

    He conducted his MSc Thesis entitled ‚ÄúLearning Partial Differential Equations from Data with Machine Learning algorithms” under the supervision of Prof. Constantinos Siettos. The work was focused on the use of machine learning, namely artificial neural networks and Gaussian processes, for the reconstruction of the FitzHugh-Nagumo PDE, from data generated at the mesoscopic level using Boltzmann-lattice modelling.

    His research interests concern numerical analysis, machine (deep) learning and mathematical modeling with particular emphasis on problems that arise in complex systems.

    Machine learning for the Modelling and Numerical Analysis of Emergent Dynamics of Phase Field Models

    Supervisor: Constantinos Siettos, Dept. of Mathematics and Applications, Università degli Studi di Napoli Federico II
    Co-supervisors: Yannis Kevrekidis, Dept. of Chemical and Biological Engineering, Johns Hopkins University

    Understanding, modelling and forecasting the emergent dynamics of complex systems from large scale spatio-temporal data is nowadays one of the most challenging open problems in the field. The hypothesis that underpins the study of complex systems is that in principle a low-dimensional description of the emergent dynamics is possible by a few macroscopic observables (e.g. density, concentration, etc.). However, for many real-world complex systems such equations in a closed form, i.e. good macroscopic descriptions, are in general unavailable. Furthermore, even the “correct” macroscopic variables for the description of the emergent behaviour may be also not known a priori. An analytical derivation of such macroscopic equations in a closed form relies on a deep understanding/intuition and prior knowledge about the physical system and assumptions that may introduce certain biases in both modelling and analysis. Therefore, the reconstruction of “accurate” coarse-grained dynamical models from data is becoming a fundamental problem in complex systems theory and its various applications. Recent developments at the junction between numerical analysis and machine learning have revolutionized the way we think about modeling and analysis from big data, thus opening the way to a new direction, where the data themselves constitute the focus of interest and the generation of hypotheses arises from data mining and analysis.

    One of the main goals of the proposed project is to set up a multi-scale equation-free/variable-free framework for the systematic extraction of coarse-scale observables from microscopic/fine-scale data, the construction and also the numerical solution of effective PDEs, using and developing new machine learning and numerical analysis algorithms. The data may come from experiments or from high dimensional microscopic simulations such as Monte-Carlo, Agent-Based and Molecular Dynamics. These detailed models, keeping track of the interactions between huge numbers of microscopic level degrees of freedom, typically lead to prohibitive computational costs for large-scale spatiotemporal simulations. Therefore, the first fundamental step of this framework is to extract the dominant features from the finer-scale data to uncover the right coarse scale observables, i.e. to find the coarse-grained manifold where the emergent dynamics effectively evolves. This is a crucial step for the development of reduced-order models that can lead to significant computational savings in large-scale spatiotemporal simulations but also to the systematic analysis (e.g. the construction of bifurcation diagrams) of the emergent dynamics. Once the right observables have been extracted/learned or even chosen with the aid of the numerical analysis if there are some physical insight about them, it is possible to learn the equations that governs the macroscopic dynamics on the manifold using traditional numerical analysis algorithms for function approximation and several machine learning methods, such as Forward and recurrent Neural Networks, Deep Learning, Random Projection Networks and Bayesian non parametric models including Gaussian Processes.

    The main applications of this project will be Phase Field Models (PFMs), that are a powerful and versatile tool for studying free-boundary complex interface problems, which can be applied to very different phenomena that give rise to very complex emergent dynamics and patterns such as solidification, cellular migration, tumor growth, mesenchymal cells, and Saffman-Taylor instability. These complex and fascinating behaviors are characterized by critical transitions that occur when small changes in inputs cause sudden, large, and often irreversible changes in the state of a system. This aspect is crucial because it highlights the unpredictable emergent dynamics of complex systems. Hence, for such complex systems, the main challenge will be to discover the coarse-scale macroscopic PDEs, from which is possible to compute the coarse-bifurcation diagrams and also construct the effective Fokker-Planck equation that will be used to assess the probability of phase-transitions near criticalities. Furthermore, this methodology will allow also to study many natural complex systems ranging from ecosystems to molecular biosystems that are known to exhibit critical transitions in their response to stochastic perturbations. Finally, this novel methodology has the potential to be applied to several systems, giving rise to disruptive innovations, e.g., for forecasting disease onset, such as tumor growth, all in all by constructing early-warning signals.