The program is structured around three thematic study areas, covering a total of 11 courses. For each course, its description with lecture hours is given below.
1. Numerical Methods (21 hours)

Course 1.1: Numerical Methods for Transport Phenomena: From Diffusion to Nonlinear Waves (6 hours)

Instructor:

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

Course Description:

In these lectures we provide an introduction to the important classes of numerical methods for partial differential equations including finite difference, finite volume and Fourier-based spectral method. After fundamentals of numerical approximation are established, some current efficient approaches are presented. The emphasis is on a solid understanding of the accuracy of these methods. Computer implementations will be demonstrated in Python. The class is suitable for graduate students from all disciplines who have interest in computational PDEs.
Prerequisite: PDEs, Fourier Analysis, Linear Algebra
Contents:

• Finite Difference Method
  • 1D Diffusion Equation, Boundary Conditions
  • Explicit and Implicit Methods for ODEs
  • Stability: von Neumann analysis, Lax Equivalence Theorem
  • CFL Condition, Super Time-Stepping (STS), Iterative Procedures
• Finite Volume Method
  • First Order Hyperbolic Conservation Laws
  • Linear Advection, Upwind Scheme, CFL Condition
  • Nonlinear Advection, Advection Diffusion Equation, Peclet Number
• Spectral Method
  • nonlinear waves (advection-diffussion-dispersion): KdV, NLS, CGL

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

1. LeVeque, R. J., Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007
2. LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002
3. Boyd, J.P., Chebyshev and Fourier Spectral Methods, Dover, 2000
4. Trefethen, L. N., Spectral Methods in Matlab, SIAM, 2000

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Course 1.2: Modeling and Simulations of Granular Flows (6 hours)

Instructor:

• Dr. Sudarshan Tiwari , RPTU Kaiserslautern - Landau, Germany

Course Description:

In this course, we have considered two main parts. The first part focuses on the modelling of granular flows, and the second part deals with numerical methods.
In the first part, we present a hierarchy of models for granular flow problems, ranging from microscopic to macroscopic models. This constitutes a multiscale modelling approach.
We begin with a microscopic model for granular flow, represented by an interacting particle system that is similar to the Discrete Element Method (DEM). We then derive a Vlasov-type equation as the limiting equation when the number of particles tends to infinity. From the Vlasov-type equation, we compute the moments and derive macroscopic equations, such as the continuity and momentum equations.
In the second part, we develop numerical methods for solving these macroscopic equations. We employ a mesh-free Lagrangian particle method, which is well suited for granular flow simulations. In this approach, the computational domain is approximated by a set of discrete moving grid points, called particles. These particles move according to their velocities and carry all relevant information, such as density and velocity. Spatial derivatives are approximated using the weighted least-squares method, which is a generalized finite difference method.
Finally, we implement the numerical schemes in Python or MATLAB to simulate granular flow problems. We investigate applications such as landslides, particle segregation, the collapse of granular columns, and other related phenomena.

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

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Course 1.3: Blood Flow Simulation using Meshfree Methods with Python (9 hours)

Instructor:

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

Course Description:

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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2: Mathematical Models & Analysis (21 hours)

Course 2.1: Moving Boundary Value Problems: FEM and Random Walk Methods (6 hours)

Instructor:

• Dr. Surendra Nepal, Linnaeus University, Sweden
  

Course Description:

In this course, we have considered two main parts. The first part focuses on the mathematical modelling and analysis of moving boundary problems arising in diffusion processes, while the second part deals with deterministic and stochastic numerical methods for their approximation.
In the first part, we present the formulation of moving boundary problems describing diffusant penetration into rubber-like materials. We begin with the physical motivation behind the model and derive a coupled system consisting of a diffusion equation and an ordinary differential equation governing the evolution of the moving interface. For the numerical treatment, we introduce a domain transformation that maps the moving domain onto a fixed reference domain. We then study the resulting transformed equations and discuss classical examples of moving boundary problems, including the Stefan problem, which models phase-change phenomena such as ice melting.
In the second part, we develop numerical methods for solving the transformed moving boundary problem. We employ the finite element method to approximate the concentration profile and the evolution of the moving interface. Furthermore, we introduce stochastic approaches based on random walk methods and establish their connection to diffusion equations. In this framework, the concentration field is represented by a collection of randomly moving particles whose dynamics are designed to capture both the diffusion process and the motion of the moving boundary. Appropriate treatments of the boundary conditions and interface evolution are incorporated into the random walk algorithm to approximate the diffusant profile and the position of moving boundary.
Finally, we implement the finite element and random walk schemes in Python and perform computational experiments to simulate diffusant penetration into rubber. The numerical results for the concentration profile and the position of the moving boundary are compared with experimental measurements.
Main resource: Lecture notes prepared by the instructor

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Course 2.2: Spectrum of the Laplace Operator (3 hours)

Instructor:

• Dr. Dhruba R. Adhikari, Kennesaw State University, USA

Course Description:

This course introduces the variational and spectral foundations of eigenvalue problems, beginning with symmetric matrices with real entries and extending to compact self-adjoint operators in infinite dimensional Hilbert spaces. The first part of the course develops the relationship between quadratic forms, the Rayleigh quotient, and eigenvalues through constrained optimization on the unit sphere. The Courant--Fischer min--max and max-min principles are discussed, providing variational characterizations of all eigenvalues of a symmetric matrix. The second part generalizes these ideas to compact self-adjoint operators on infinite dimensional real Hilbert spaces, proving the existence of eigenvalues and eigenvectors, orthogonality of eigenspaces, and the spectral theorem. These results are then applied to the Laplace operator, leading to the study of the standard Dirichlet eigenvalue problem. Particular emphasis is placed on variational characterizations and qualitative properties of the first eigenvalue.

Main resource: Lecture notes prepared by the instructor
Supplementary resources:

• Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2010
• Djairo Guedes de Figueiredo, Positive Solutions of Semilinear Elliptic Problems, Lecture Notes in Mathematics, Proceedings of the 1st Latin American School of Differential Equations, Sao Paulo, Brazil, June 29-July 17, 1981
• David C. Lay, Steven R. Lay and Judi J. McDonald, Linear Algebra and its Applications, Fifth Edition, Pearson, 2016
• Dragisa Mitrovic and Darko Zubrinic, Fundamentals of Applied Functional Analysis: Distributions, Sobolev Spaces, Nonlinear Elliptic Equations, Pitman Monographs and Surveys in Pure and Applied Mathematics, 91, Longman, 1998

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Course 2.3: Introduction to Sobolev Spaces (6 hours)

Instructor:

• Dr. Rosa Pardo, Universidad Complutense de Madrid, Spain
  

Course Description:

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

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

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Course 2.4: Calculus of Variations with Applications to PDEs (6 hours)

Instructor:

• Dr. Maria Mabel Cuesta León, University of the Littoral Opal Coast ULCO, France
  

Course Description:

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:

• 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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3. Data-Driven Approaches (18 hours)

Course 3.1: Data Mining and Machine Learning Models (6 hours)

Instructors:

• Dr. Gokarna Aryal, Purdue University Northwest, USA
• Dr. Ram C. Kafle, Sam Houston State University, USA

Course Description:

This course provides a comprehensive introduction to the fundamental principles and commonly used methodologies in data mining and machine learning. Participants will learn how to extract knowledge from data, develop predictive models, and apply modern machine learning methods to problems arising across a wide range of scientific disciplines.
The course begins with basic supervised learning techniques, including linear regression and logistic regression, which serve as the foundation for predictive modeling and statistical learning. Building upon these fundamentals, participants will explore key concepts that are essential for real-world data analysis, such as data preprocessing, visualization, handling missing values, addressing class imbalance, model evaluation, and cross-validation strategies. More advanced machine learning methods including support vector machines, regression and decision trees, and regularization techniques such as lasso regression for feature selection and model simplification will be discussed. Participants will explore ensemble learning approaches, with particular emphasis on bagging and boosting methods and learn how aggregating multiple models can lead to improved predictive performance and increased model stability.
Throughout the course, theoretical concepts will be complemented by practical examples and hands-on applications using statistical software R, enabling participants to gain experience in selecting appropriate algorithms, training and validating models, interpreting results, and assessing model performance. Designed for graduate students, advanced undergraduate students, and early-career researchers from a broad range of disciplines, this course provides a solid foundation for further study and research in data-driven modeling, machine learning, and scientific data analysis. By the end of the course, participants will have acquired the essential knowledge and practical skills required to apply modern machine learning methods to data-driven modeling and scientific research.
Main resource: Lecture notes and computational labs prepared by the instructors

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Course 3.2: Machine Learning and Artificial Intelligence for Spatial Data Analysis using QGIS and Python (3 hours)

Instructor:

• Dr. Bidur Devkota, Gandaki College of Engineering and Science, Pokhara University, Nepal

Course Description:

This hands-on course introduces Geographic Information Systems (GIS) and their integration with machine learning for real-world applications. Using open-source tools (QGIS and Python), participants learn practical skills through case studies in urban planning and landslide risk assessment.
The course is organized into five modules:
Module 1: Introduction to GIS
Covers core GIS concepts, the GIS workflow, and the relationship between GIS and Remote Sensing. Explores application domains including disaster management, urban planning, and public health.
Module 2: Introduction to QGIS
Practical training on QGIS, including installation, interface navigation, adding various data formats (Shapefile, GeoJSON, GeoTIFF), understanding Coordinate Reference Systems, and basic styling and symbology.
Module 3: Spatial Operations in QGIS: River Buffer Risk Analysis and Building Density Mapping
Hands-on case study in Kathmandu using spatial queries (Extract by Location, Difference) to identify buildings in flood risk zones. Includes distance calculations, ward-level statistics (building counts, density per hectare), and visualizations like heatmaps and fishnet grids.
Module 4: Geospatial Machine Learning: Predicting Nighttime Lights from Building Types
Bridges GIS and data science by training a linear regression model to predict Nighttime Light intensity based on building types. Identifies "light deficit" wards needing infrastructure investment. Provides a baseline model, with suggestions to improve performance using additional features (e.g., distance to roads, building area) and advanced models like Random Forest or XGBoost.
Module 5: Landslide Risk Analysis with Python
Advanced Python-based workflow integrating multiple factors: Soil Erodibility (K-factor) using the Panagos (2014) method, Water Table Depth estimation, and Weighted Overlay Analysis. Produces a continuous landslide risk index classified into standard categories (Very Low to Very High) for hazard mapping and planning.

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

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Course 3.3: Deep Learning Techniques for Computer Vision (3 hours)

Instructor:

• Dr. Ramchandra Rimal, Middle Tennessee State University, USA

Course Description:

This course introduces the core paradigms of machine learning, beginning with the formulation of predictive modeling problems. We establish foundational principles using a single-neuron model and explore various activation functions to illustrate how this architecture relates to logistic regression and other traditional machine learning algorithms. Building upon this, the curriculum extends to Multi-Layer Perceptrons (MLPs), detailing their capacity as universal approximators and examining backpropagation as the algorithmic backbone of the learning process. However, when applied to computer vision tasks, MLPs encounter severe limitations with high-dimensional, spatially structured inputs, such as excessive parameterization and a lack of translation invariance. To overcome these challenges, the course introduces Convolutional Neural Networks (CNNs), diving deep into their underlying mechanisms, including local receptive fields, weight sharing, and pooling operations, as principled solutions for spatial feature extraction.
Reinforcing conceptual intuition with practical application, the course features a comprehensive, hands-on learning segment using Python. Participants will explore the complete deep learning workflow using the open-source iNaturalist dataset for a multiclass classification problem. This practical demonstration covers the entire modeling pipeline, from robust data ingestion and augmentation to model validation and comprehensive results analysis. A key focus is placed on transfer learning, exploring the mechanisms of adapting pre-trained networks, and identifying the specific scenarios where this approach is highly advantageous. Furthermore, participants will learn the principles of designing custom CNN architectures from scratch and how to appropriately tune both custom and fine-tuned models using regularization strategies and validation-driven early stopping.
Building on this technical foundation, the course will survey impactful scientific applications of modern vision systems. We emphasize rigorous dataset curation and diagnostic evaluation, demonstrating how to critically analyze model performance using loss plots, confusion matrices, and standard performance metrics for classification modeling. Tailored for an interdisciplinary STEM audience, this course bridges the gap between theoretical foundations and applied modeling, enabling participants to rigorously evaluate and integrate deep learning architectures into their specific scientific domains.

Main resource: Lecture notes and hands-on computational labs prepared and provided by the instructor

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Course 3.4: Machine Learning based Turbulence Prediction in Urban Areas (6 hours)

Instructor:

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

Course Description:

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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