Galaris, Evangelos
(2022)
Physics-Informed Numerical Analysis and Machine Learning for Modelling and Analysis of Complex Systems with Applications in Computational Neuroscience.
[Tesi di dottorato]
| Tipologia del documento: |
Tesi di dottorato
|
| Lingua: |
English |
| Titolo: |
Physics-Informed Numerical Analysis and Machine Learning for Modelling and Analysis of Complex Systems with Applications in Computational Neuroscience |
| Autori: |
| Autore | Email |
|---|
| Galaris, Evangelos | evangelos.galaris@unina.it |
|
| Data: |
10 Marzo 2022 |
| Numero di pagine: |
128 |
| Istituzione: |
Università degli Studi di Napoli Federico II |
| Dipartimento: |
Matematica e Applicazioni "Renato Caccioppoli" |
| Dottorato: |
Matematica e Applicazioni |
| Ciclo di dottorato: |
34 |
| Coordinatore del Corso di dottorato: |
| nome | email |
|---|
| Moscariello, Gioconda | gioconda.moscariello@unina.it |
|
| Tutor: |
| nome | email |
|---|
| Siettos, Constantinos | [non definito] |
|
| Data: |
10 Marzo 2022 |
| Numero di pagine: |
128 |
| Parole chiave: |
physics-informed machine learning, complex systems, computational neuroscience |
| Settori scientifico-disciplinari del MIUR: |
Area 01 - Scienze matematiche e informatiche > MAT/08 - Analisi numerica |
[error in script]
[error in script]
| Depositato il: |
24 Mar 2022 07:12 |
| Ultima modifica: |
28 Feb 2024 10:49 |
| URI: |
http://www.fedoa.unina.it/id/eprint/14443 |
Abstract
This Ph.D. Thesis deals with the development and implementation of physics-informed numerical analysis and machine learning methods for the modelling and bifurcation analysis of large-scale complex dynamical systems from big data as well as the numerical solution of low-dimensional models of Ordinary Differential Equations (ODEs). Modelling and analysing the emergent behaviour of large-scale complex dynamical systems from experimental data and/or data produced by detailed high-fidelity microscopic simulations, requires appropriate data-driven/numerical analysis-based methods for extracting coarse-scale models that can be utilized for further numerical analysis of the emergent dynamics.
The springs of this Thesis are interdisciplinary, bridging state-of-the-art methodologies from numerical analysis, machine learning, microscopic simulations, bifurcation theory and neuroimaging. The research efforts and results were focused in three main directions.
We have first focused on the solution of the source localization problem in neuroimaging, exploiting and comparing the performance of state-of-the-art regularization methods, namely the standarized Low Resolution Electromagnetic Tomography (sLORETA), the weighted Minimum Norm Estimation (wMNE) and the dynamic Statistical Parametric Mapping (dSPM) and information arising from the electrophysiology of the brain. In particular, the research efforts were focused on the localization of the sources of brain activity of children with epilepsy based on electroencephalograph (EEG) recordings acquired during a visual discrimination working memory (WM) task using numerical regularization algorithms. Importantly, our study and findings reveal also the importance and potential that originates from the use of physics-based information as well as publicly available scientific resources such as the "Neurodevelopmental MRI" database, which allow the researchers to numerically analyse available neuroimaging data and investigate questions beyond the scope of the original studies.
Next, we addressed a computational framework for the embedding of high-dimensional spatio-temporal data produced by microscopic simulators in low-dimensional manifolds, the identification of the appropriate parsimonious observables based on the constructed low-dimensional models and finally the construction of coarse-grained bifurcation diagrams from spatio-temporal data. Thus, we exploit manifold learning and in particular Diffusion Maps to identify the intrinsic dimension of the manifold where the emergent dynamics evolve and for feature selection. Based on the selected features, we learn the right-hand-side of the effective partial differential equations (PDEs) using Feed-forward Neural Networks (FNNs). Based on the learned black-box model, we construct the corresponding bifurcation diagrams, exploiting the arsenal of numerical bifurcation theory. For our illustrations, we implemented the proposed method to construct the one dimensional bifurcation diagram of the celebrated and well-studied FitzHugh-Nagumo PDEs of activation-inhibition dynamics in neurons from data generated by Lattice Boltzmann simulations.
Finally, we addressed a numerical method based on physics-informed random-projection neural networks for the solution of initial value problems (IVPs) of low-dimensional systems of ODEs with a focus on stiff problems. The numerical solution of the IVPs is obtained by constructing a system of nonlinear algebraic equations, which is solved with respect to the output weights by the Gauss-Newton method. The performance of the proposed scheme was assessed through three benchmark stiff IVPs, namely the Prothero-Robinson, the van der Pol model and the the ROBER problem. Furthermore, the proposed scheme was compared with an adaptive Runge-Kutta method, and a variable-step variable-order multistep solver based on numerical differentiation formulas, as implemented in the ode45 and ode15s of the matlab ODE suite. We show that the proposed scheme yields good approximation accuracy, thus outperforming in some cases ode45 and ode15s, especially in the cases where steep gradients arise. Furthermore, the computational times of our approach are comparable with those of the two matlab solvers for all practical purposes.
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