### Overview

The lectures will be given by Professor Adil Rasheed and Professor Trond Kvamsdal, both from the Norwegian University of Technology and Science. The aim of the lecture is to introduce the field of Hybrid Analysis and Modeling (HAM) in the context of fluid mechanics. Until recently, pure physics based modelling approach were employed to solve complex problems in the field of fluid mechanics and use of data-driven models were limited to postprocessing or parameterization development. With the rise of deep learning new ways of combing the two approaches are emerging. In the context of this course we call this approach Hybrid Analysis and Modeling (HAM). The approach combines the strengths of both the approach while eliminating their weaknesses.

Key questions that will be addresses are:

1. What are the strengths and weaknesses of physics-based and data-driven modelling?

2. How to discover equations from data directly without invoking first principle?

3. How to speed up numerical simulation using Reduced Order Modeling?

4. How to eliminate some of the weaknesses of Reduced Order Modeling, such as closure problems, stability and unknown physics.

**Tentative content:**

**Lecture 1: Introduction to Machine Learning**

- ML methods relevant for HAM
- Unsupervised ML: PCA
- Supervised ML: Deep Learning, Sparse Regression
- Symbolic regression

- Strengths and weaknesses of these methods

**Lecture 2: Reduced Order Modeling**

- Proper Orthogonal Decomposition
- Galerkin Projection based ROM
- Closure problems
- Snapshot generations

**Lecture 3: Hybrid Analysis and Modeling**

- Data-driven equation discovery
- Non-intrusive ROM
- Physics guided ML
- Physics informed ML

**Teaching material:**

- Python notebooks
- A collection of recent journal papers
- Quarteroni A, Manzoni A and Negri F: Reduced basis methods for partial differential equations. Springer International Publishing, 2016.