01DTTNG

A.A. 2023/24

Course Language

Inglese

Course degree

Master of science-level of the Bologna process in Ingegneria Matematica - Torino

Course structure

Teaching | Hours |
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Teachers

Teacher | Status | SSD | h.Les | h.Ex | h.Lab | h.Tut | Years teaching |
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Teaching assistant

Context

SSD | CFU | Activities | Area context |
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MAT/08 | 6 | D - A scelta dello studente | A scelta dello studente |

2022/23

The course is an introduction to some general methodologies for the numerical treatment of partial differential equations modeling phenomena of engineering interest.
The course will focus on Finite Elements, Spectral Methods, and Finite Volume methods. Iterative methods for large linear and non-linear systems will be considered.
The most relevant results from the theoretical point of view will be rigorously proven when the proofs are based on mathematical arguments of wide applicability.
The exercise lesson will introduce the use of mathematical software to solve problems governed by partial differential equations.

Complex phenomena in different branches of human knowledge (such as e.g. Physics, Engineering, Biological and Social Sciences) can be simulated by sophisticated numerical methods, providing a high-fidelity description of reality. However, often these simulation techniques turn out to be expensive or slow, particularly when responses are needed in large numbers or in real-time.
Reduced order methods have been devised to surrogate the accurate but expensive simulation models, by producing results that are accurate enough but cheaper to obtain. Proper Orthogonal Decompositions (POD), or Reduced Basis Methods (RBM) are examples of such techniques. More recently, the use of Machine Learning (ML) techniques has been advocated as a viable alternative to more classical strategies, particularly in the construction of nonlinear models.
The course is an introduction to Reduced-Order Methods on the one hand, and to Machine Learning on the other hand, both in their theoretical aspects and in their practical implementation. They will be combined to produce efficient solution strategies, which will be applied to treat representative mathematical models described by partial differential equations. Applications will range from parametric boundary-value problems to uncertainty quantification or optimal control problems.
The course provides essential knowledge on surrogate simulation models, which students could exploit in many scientific and industrial contexts during their future careers.

At the end of the course, students should:
- know the basic principles of reduced-order models
- know the basic principles of machine learning
- know some specific software for reduced-order models and machine learning
- be able to handle a POD technique
- be able to handle a greedy algorithm
- be able to design and run a reduced-basis technique to solve a simple parametric problem
- be able to handle a feed-forward neural network
- be able to design and run a Physics informed neural network to solve a simple parametric problem
Altogether, this set of knowledge and skills should allow the student, in a future professional environment, to make appropriate choices about the reduction of complexity in numerical modelling and simulation.

At the end of the course, students should:
- know the basic principles of Reduced-Order Models
- know the basic principles of Machine Learning
- know the main algorithms and structures of reduced-order models and machine learning
- know some specific software for reduced-order models and machine learning
- be able to create reduced-order models and neural networks based on existing open-source platforms
- be able to handle a POD technique
- be able to handle a greedy algorithm
- be able to design and run a reduced-basis technique to solve a simple parametric problem
- be able to handle a feed-forward neural network
- be able to design and run a Physics Informed Neural Network (PINN) to solve simple parametric problems.
Altogether, this set of knowledge and skills should allow the student, in a future professional environment, to make appropriate choices about the reduction of complexity in numerical modelling and simulation.

Basic notions about numerical methods and programming (in Matlab, C, C++, or Python).
Some knowledge of numerical methods for partial differential equations (such as the finite element method) is helpful, but not strictly required.

Basic notions about numerical methods and programming (in C++ or Python) are recommended.
Some knowledge of numerical methods for partial differential equations (such as the finite element method) is helpful, but not strictly required.

- Introduction to reduced-order models (5h)
- The Proper Orthogonal Decomposition (POD) (5h)
- Greedy algorithms (5h)
- Reduced basis methods (10h)
- Empirical interpolation methods (3h)
- Introduction to feed-forward neural networks (NNs) (9h)
- Introduction to Python and Tensorflow (6h)
- The POD-NN technique (3h)
- Physics-informed neural network (PINNs) (6h)
- Engineering applications (8h)

- Introduction to reduced-order models (5h)
- The Proper Orthogonal Decomposition (POD) (5h)
- Greedy algorithms (5h)
- Reduced Basis Methods (10h)
- Empirical interpolation methods (3h)
- Introduction to feed-forward neural networks (NNs) (9h)
- Introduction to Python and Tensorflow (6h)
- The POD-NN technique (3h)
- Physics-informed neural network (PINNs) (6h)
- Engineering applications (8h)

The course is organized as follows:
- 30 hours of class lessons, where the basic concepts of reduced order methods and of machine learning will be given, and the students will learn how to integrate such concepts in view of the development of efficient computational tools.
- 30 hours of exercises, partly in classroom (typically whenever an argument has been completed at lesson) and partly in laboratory, to develop codes and apply them to the solution of practical problems.

The course is organized as follows:
- 30 hours of class lessons, where the basic concepts of reduced-order methods and of machine learning will be given, and the students will learn how to integrate such concepts in view of the development of efficient computational tools.
- 30 hours of exercises, partly in a classroom (typically whenever an argument has been completed at lesson) and partly in a laboratory, to develop codes and apply them to the solution of practical problems.

Electronic material provided by the teachers.
Reference books:
- J. Hesthaven, G. Rozza, B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Springer 2016
- T.M. Mitchell, Machine Learning, McGraw-Hill 2019

Electronic material provided by the teachers.
Reference books:
- J. Hesthaven, G. Rozza, B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Springer 2016
- T.M. Mitchell, Machine Learning, McGraw-Hill 2019

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.

The oral exam consists of two questions on the theoretical aspects, and a discussion of the project.
The weight of the two parts in the formation of the vote is approximately equal. The exam lasts between 30 and 45 minutes.
The evaluations are expressed out of 30 and the exam is passed if the score reported is at least 18/30. The maximum rating is 30 with honors (30 lode).
During the examination concerning the theoretical questions, it is not allowed to keep any source of information.
During the discussion of the project, the student can make use of it and of the codes developed for its drafting.

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.

© Politecnico di Torino

Corso Duca degli Abruzzi, 24 - 10129 Torino, ITALY

Corso Duca degli Abruzzi, 24 - 10129 Torino, ITALY