The course consists of two modules Introduction to data science [4 credits] This short course will give an introduction to data science for physicists. The course will introduce the basics of data science problems, and introduce students to Bayesian techniques for inference, unsupervised learning methods for exploratory data analysis, and neural networks. On the theoretical side, students will learn fundamentals of random matrix theory and of the theory of neural networks. Advanced numerical methods [4 credits] Introduction to numerical methods applied to molecular dynamics simulations. The course covers the theoretical foundations of molecular dynamics and Monte Carlo simulations, starting from fundamental concepts from Newtonian dynamics, statistical mechanics, and variational principles. The course includes practical sessions where students implementing and applying the algorithms.

Introduction to data science After this course, students will 1. Understand the fundamental ingredients of a data science problems 2. Be able to apply the basics of Bayesian inference problems to inference problems 3. Know the basics of random matrix theory, including the Wigner and Wishart ensembles as well as their spectra 4. Know basic unsupervised learning techniques 5. Know the basics of neural network architectures and neural learning dynamics in supervised learning. Advanced numerical methods The student will be able to understand the basic concepts of statistical mechanics, and will be capable of executing molecular dynamics simulations in different ensembles.

Introduction to data science Pre-requisites are probability, linear algebra and analysis at the level obtained after successful completion of a Bachelor degree in Physics, Mathematics, Mathematical Engineering, Computer science, or a similar course. The course does not assume any knowledge of data science. Advanced numerical methods Basic knowledge of the Python programming language. Basic concepts in mechanics (equations of motion, Poisson brackets) and calculus.

Introduction to data science I. Basics of Bayesian inference [R. Trotta] Introduction to inference: what is it? Why do we need it? Frequentist probability vs bayesian probability Confidence levels, posterior distributions, priors and the difference between all those If time allows: classical hypothesis testing vs Bayesian model comparison II. An introduction to random matrix theory [J. Barbier] The Wigner ensemble and the semi-circular law Wishart Ensemble and Marchenko–Pastur law Spiked matrix models and the BBP transition III. Introduction to unsupervised learning and dimensional reduction [A. Laio] Principal Component Analysis Multidimensional scaling and kernel methods Intrinsic dimension estimates IV. Neural networks [S. Goldt] Neural networks 101: types and applications + some open theoretical problems Learning dynamics: empirical phenomena and theoretical predictions The impact of data structure Unsupervised learning: from Hebbian learning to independent components Advanced numerical methods (G.Bussi and C. Micheletti) 1.Basic concepts of Newtonian dynamics and Statistical Mechanics: energy conservation, time reversibility and phase-space incompressibility, Liouville Theorem, Ergodicity. Derivation of equilibrium statistical mechanics from variational principles (definition of entropy, ideal gas, microcanonical, canonical and grandcanonical statistical ensembles, law of mass acrtion) [10h] 2.Integration schemes for molecular dynamics: Verlet, Trotter splitting, Velocity Verlet. Dependence of the results on the time step. [10h] 3.Sampling the canonical ensemble with Monte Carlo: Metropolis-Hastings rule, balance and detailed balance, hybrid Monte Carlo. [10h] 4.Sampling the canonical ensemble with molecular dynamics: velocity rescaling, Berendsen thermostat, Andersen thermostat, Langevin dynamics, stochastic velocity rescaling. Stochastic equations: Itoh rule, Fokker-Planck equation. [10h]

STUDY MATERIALS: Introduction to data science A set of papers and Python notebooks accompany the course Advanced numerical methods for topics 2, 3, and 4: Recording of the lectures, Handouts, Python notebook with solutions of the exercises

Introduction to data science This is an entirely taught course. Advanced numerical methods 30h are devoted to theory classes. 10h are devoted to the correction of computational exercises done independently by the student at home.

Introduction to data science As an introduction to the course, chapters 1-3, 20-21, 38-39, 41, and 44 of the book “Information Theory, Inference, and Learning Algorithms” by David MacKay are recommended reading. Advanced numerical methods Kerson Huang; Statistical mechanics Julia Yeomans; Statistical mechanics of Phase transitions Frenkel – Smith; Understanding Molecular Simulation: From Algorithms to Applications Tuckerman; Statistical Mechanics: Theory and Molecular Simulation

Esercizi risolti; Video lezioni dell’anno corrente;

Modalità di esame: Prova scritta (in aula); Prova orale obbligatoria;

Exam: Written test; Compulsory oral exam;

... Introduction to data science The course will be assessed by a written exam of two hours, covering all four topics of the course. Advanced numerical methods Oral exam.

Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.