The program is structured around three thematic study courses, covering a total of 12 content topics/sections. The course description with lecture hours breakdwon for each course is given below.

Course 1: Mathematical Models & Analysis (18-24 hrs)

Section 1.1: Introduction to Sobolev Spaces and Calculus of Variations with Applications to PDEs

Instructors:

• Dr. Mabel Cuesta León, University of the Littoral Opal Coast ULCO, France
  
• Dr. Rosa Pardo, Universidad Complutense de Madrid, Spain
  

Abstract:

Part I: An introduction to Sobolev spaces
Sobolev spaces are widely used in the study of Partial Differential Equations. The aim of this course is to give the basic properties of these espaces and to apply them to some elementary PDE and ODE boundary problems.

Contents:

• Weak derivatives. Examples.
• The spaces H¹ and H₀¹. The 1-dimensional case
• Poincaré inequality
• Sobolev embeddings
• Theorem of Rellich-Kondratov
• Traces

Part II: Calculus of Variations with applications to PDEs
The Calculus of Variations identifies an important class of nonlinear PDE that can be solved using relatively simple ideas from nonlinear Functional Analysis. The Calculus of Variations relates one PDE with the critical points of some 'energy' functional, and usually this is an easier problem to be solved. In addition, many of the laws of physics and other scientific disciplines arise directly from the minimum of the energy functional, in other words as variational problems.

Contents:

• Introduction. Principle of least action. Examples: minimum surface of revolution. Brachistochrone.
• Calculus of variations for PDEs
• Existence of minimizers
• Existence of weak solutions to the Euler-Lagrange equation
• Variational problems with constraints. Lagrange multipliers
• Theorem of Rellich-Kondratov
• Critical points. The Mountain Pass lemma

Main resource:Lecture notes and worksheets prepared by the instructors
Supplementary resources:

• Brezis, Haim: Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. xiv+599 pp.
  ISBN: 978-0-387-70913-0
• Dacorogna, Bernard: Introduction to the calculus of variations. Third edition. Imperial College Press, London, 2015. x+311 pp. ISBN: 978-1-78326-551-0
• Struwe, Michael: Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 34.Springer-Verlag, Berlin, 2008. xx+302 pp.
  ISBN: 978-3-540-74012-4

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Section 1.2: Partial Differential Equations: Theory of Water Waves

Instructor:

• Dr. Stefani Mancas, Laboratory for Physical Sciences (LPS), University of Maryland, College Park, MD, USA

Abstract:

In these lectures we start with techniques of 19th century mathematics that are brought alive to deal with linear waves theory. We then continue with 20th century nonlinear water waves in which we will look at KdV, Burgers and NLS equations. By the end of the short course students should get a basic understanding on the theory of water waves and will be ready to apply it to several real-world nonlinear models that include traffic flow, flood waves, chromatographic models, sediment transport in rivers, glacier flow, and roll waves.

Main resource: Lecture notes and problem sets prepared by the instructor
Supplementary resources:

• Drazin, Philip G. and Johnson, Robin Stanley Johnson. Solitons: an introduction. Vol. 2. Cambridge University Press, 1989.
• Debnath, Lokenath. Nonlinear partial differential equations for scientists and engineers. Boston: Birkhäuser, 2005.
• Billingham, John and King, Andrew C.. Wave motion. No. 24. Cambridge University Press, 2000.

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Section 1.3: Mathematical Methods for Nonlinear Partial Differential Equations: Classical Theory and Some New Trends

Instructor:

• Dr. Andrei Ludu, Embry-Riddle Aeronautical University, Daytona Beach, USA

Abstract:

This lecture aims at introducing PhD students and young researchers to the classical theories and methods for the analysis and solving of some important nonlinear partial differential equations (NLPDE). We will introduce the concept of integrability for NLPDE from different mathematical and theoretical physics points of view, including differential algebra (local conservation laws, Bäcklund–Darboux transforms), algebraic geometry (Theta and Baker functions), and the inverse scattering method (Riemann–Hilbert problem). We will also present certain symplectic, Lie theoretic, and differential geometric properties of the soliton solutions for these NLPDE. We will discuss elements of the Kolmogorov-Arnold-Moser (KAM) theory which plays an important role in our understanding of the behavior of non-integrable and almost integrable Hamiltonian systems. Applications to various physics equations including the traditional KdV, MKDV, sine–Gordon, Nonlinear Schrödinger equations, also dispersive waves, nervous signals, bio-membranes will be described.

Main resource: Lecture notes and worksheets prepared by the instructor

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Section 1.4: Topological solitons: kinks and vortices

Instructor:

• Dr. Maciej Dunajski, University of Cambridge, UK

Abstract:

Most nonlinear differential equations arising in natural sciences admit chaotic behavior and cannot be solved analytically. Integrable systems lie on the other extreme. They possess regular, stable, and well-behaved solutions known as solitons. These solutions play important roles in pure and applied mathematics as well as in theoretical physics where they describe configurations topologically different from vacuum. I will introduce two classes of such solitons in 1+1 and 2+1 dimensions: scalar kinks, and abelian vortices.

Main resource: Lecture notes and worksheets prepared by the instructors
Supplementary resources: Solitons, Instantons, and Twistors by Maciej Dunajsky, 2nd edition, Oxford University Press, 2024,
https://global.oup.com/academic/product/solitons-instantons-and-twistors-9780198872542?cc=it&lang=en&

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Course 2: Numerical Methods (18-24 hrs)

Section 2.1: Introduction to Python for Scientific Computing

Instructor:

• Dr. Harihar Khanal, Embry-Riddle Aeronautical University, USA

Abstract:

This course aims to teach programming in Python within the framework of Scientific Computing. Starting with basic Python scripts, it progressively introduces essential modules like NumPy, Matplotlib, and SciPy, which are extensively used in numerical analysis and scientific computing. Through pedagogically designed computational labs, students will explore topics such as numerical quadrature, Monte Carlo integration, solving linear algebraic systems, ODE solvers, numerical methods for PDEs, data fitting, statistical sensitivity analysis, and parameter optimization. The course emphasizes both the accuracy and performance of numerical methods, making it ideal for graduate students from various disciplines who are interested in mathematical modeling, numerical analysis, and scientific computing.

Main resource: Lecture notes and computational labs prepared by the instructor

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Section 2.2: Blood Flow Simulation using Meshfree Methods with Python

Instructor:

• Dr. Panchacharam Mariappan, Indian Institute of Technology Tirupati, India

Abstract:

Through this course, participants will learn how to simulate the blood flow using meshfree methods, and how to implement it through Python programming. Focusing on biomedical applications, the course covers fluid dynamics fundamentals, an overview of the traditional mesh-based methods and the generalized finite difference method. Hands-on Python coding sessions will guide participants through implementing mesh-free algorithms, solving real-world cardiovascular problems, and visualizing results, equipping them with the skills to tackle biomedical fluid dynamics research and development.

Main resource: Lecture notes and worksheets prepared by the instructor

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Section 2.3: Computational Fluid Mechanics

Instructor:

• Dr. Sri Redjeki Pudjaprasetya, Institut Teknologi Bandung, Indonesia

Abstract:

Fluid processes are inherently complex and analytical solutions describing fluid dynamics exist only in a few instances and under simplified assumptions. Because of its complexity, computer-based numerical models are required to approximate fluid behavior in more realistic situations. In this course, I will discuss numerical schemes based on staggered grids for solving fluid flow problems. This method is relatively simple, efficient, and robust; this method used in many commercial CFD software. During the course, numerical models are gradually built up and refined with the objective to illustrate and explore various dynamical processes occurring in fluids. Analytical solutions of certain fluid phenomena are used as invaluable benchmarks for verification of these model simulations. This course is of interest by various disciplines like applied mathematics, fluid mechanics, numerical analysis, and computational science.

Main resource: Lecture notes and computational labs prepared by the instructor

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Section 2.4: Stable Numerical Methods for Applications in Computational Fluid Dynamics

Instructor:

• Dr. Sofia Eriksson, Linnaeus University, Sweden

Abstract:

Typical applications are Initial Boundary Value Problems in different areas of computational fluid dynamics, modeled by equations such as the compressible and incompressible Navier–Stokes and Euler equations, the wave equation, the heat equation, the advection–diffusion equation and so on. These equations are approximated numerically using finite difference methods designed to satisfy summation-by-parts relations (SBP). However, Initial Boundary Value Problems also need boundary conditions, and here those are imposed using a penalty technique denoted SAT. Together, the SBP-SAT technique facilitates constructing numerical schemes that are both high order accurate and time-stable.

Main resource: Lecture notes and problem sets prepared by the instructor

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Section 2.5: Modeling and Simulations of Multi-Scale Problems in Gas Dynamics

Instructor:

• Dr. Axel Klar, RPTU Kaiserslautern - Landau, Germany

Abstract:

We derive a hierarchy of models for gas dynamics starting from microscopic models and deriving kinetic models, in particular mean-field and Boltzmann equations. We further discuss properties of these kinetic equations and derive their hydro-dynamic limits such as Diffusion, Euler or Navier-Stokes equations. Moreover, simplifications of the kinetic equation like the BGK equation are considered. In the following, numerical methods for the above equations are presented ranging from stochastic to deterministic algorithms for kinetic and fluid dynamic equations. Special focus is put-on so-called mesh-free methods based on moving least squares technologies. Such methods are discussed in the framework of Semi-Lagrangian (SL) and Arbitrary-Lagrangian-Eulerian (ALE) methods.

Main resource: Lecture notes and worksheets prepared by the instructor

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Course 3: Data-Driven Approaches (12-18 hrs)

Section 3.1: A User-Friendly Guide to Uncertainty Quantification for PDEs and ODEs

Instructor:

• Dr. Wolfgang Bock, Linnaeus University, Sweden

Abstract:

Uncertainty quantification aims at studying the impact of aleatory- (e.g. natural variability) or epistemic uncertainty onto computational models used in science and engineering. The course introduces the basic concepts of uncertainty quantification: probabilistic modeling of data, uncertainty propagation techniques (polynomial chaos expansions) and sensitivity analysis. After this course students will be able to properly define an uncertainty quantification problem, select the appropriate computational methods and interpret the results in meaningful statements for field scientists, engineers and decision makers.

Main resource: Lecture notes and problem sets prepared by the instructor

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Section 3.2: Machine Learning and Deep Learning Techniques for PDEs

Instructor:

• Dr. Prashant Shekhar, Embry-Riddle Aeronautical University, USA

Abstract:

WIth the latest developments in the field of machine Learning and deep learning, researchers are using these techniques in a wide variety of fields such as medicine, healthcare, engineering, and basic sciences. Keeping this in mind, this course aims to provide foundational understanding of the deep learning techniques for solving physics problems with multiple flavors. Starting with the basics of scientific computing in python, we will discuss packages like numpy and PyTorch that are widely used for manipulating matrices and tensors. Following this, we will take a deep dive into performing non-linear regression with deep learning models such as neural networks. Since any AI model is learning a regression function at its heart, such training will provide participants with the foundational hands-on knowledge for using AI models for physics. Following this, we will be coding Physics Informed Neural Networks (PINNs) from scratch to solve problems such as modeling a damped harmonic oscillator, and modeling Burgers equation. With our detailed discussion on training and convergence of such models, the course will provide participants with a deep intuitive understanding of leveraging deep learning for other physics-based problems. The course will conclude by summarizing general applications of PINNs and other physics driven machine learning approaches in solving problems of practical importance. With its promised deliverables, the course is expected to be useful for students, researchers, and other scientists working in all data or physics driven domains.

Main resource: Lecture notes and computational labs prepared by the instructor

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Section 3.3: Machine Learning based Turbulence Prediction in Urban Areas

Instructor:

• Dr. Maximilian Dauner, HM Hochschule München University of Applied Sciences, Germany

Abstract:

This summer school course offers participants a comprehensive introduction to the use of machine learning for predicting wind turbulence in urban areas. The course will explore the key requirements, challenges and neural network architectures involved in turbulence modeling. The course begins by addressing the data requirements for accurately predicting airflow in urban environments using neural networks. Participants will learn about the importance of training datasets, which must include high spatial and temporal resolution to capture complex eddies around building structures. It will also cover the necessary tools and equations for generating suitable datasets, along with guidance on the variety of scenarios, such as different wind directions, speeds and urban layouts, needed to train robust models capable of generalizing to new environments. Participants will then explore state-of-the-art neural network architectures designed for learning partial differential equations (PDEs), with a focus on the Fourier Neural Operator (FNO). This model can learn entire families of PDEs and produce grid-independent predictions. A live coding session in Python, using the PyTorch library, will guide participants through the development and functionality of the FNO. The course concludes with an exploration of real-world applications, showcasing results from FNO-based predictions of wind turbulence in urban areas and how this information can be used to optimize flight paths for autonomous drones. Participants will also be introduced to other cutting-edge models, including those that leverage the FNO to predict extreme global weather events.

Main resource: Lecture notes and computational labs prepared by the instructor
Supplementary resources: Zongyi Li, Nikola B. Kovachki, K. Azizzadenesheli, Burigede Liu, K. Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. In International Conference on Learning Representations. https://openreview.net/forum?id=c8P9NQVtmnO

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