# Data-driven Modeling of Conservation Laws (MOD600)

Nowadays we have access to data steaming from various transport type of processes in engineering setting (e.g., flow of fluid), daily life (e.g., traffic flow), or health care (e.g., how to deal with cancer tumors). The development of digital tools that involve a proper combination of observation data and knowledge about underlying physical laws that govern such processes, appears to be an attractive approach and is an alternative to pure machine learning based methods. This course gives an introduction to a crucial class of mathematical equations, so-called conservation laws. These equations arise from well-established physical laws that surround our lives and examples range from processes governed by external physical forces like gravity and friction and mechanisms that regulate flow of traffic, to more delicate microscopic force balances exploited by the cancer cells to make them a threat to life. The structure of such conservation laws is examined as well as how to compute exact and numerical solutions. The material is presented in terms of mandatory projects and involve formulation of such mathematical models for a given application, implementing Python scripts to solve such models numerically, as well as training of such models against observation data by means of powerful statistical method.

Course description for study year 2024-2025

Facts

MOD600

1

10

Spring

1

Spring

English

Time table

## Content

Fundamental mathematical models for studying transport in continuous media of high relevance for computational fluids engineering, earth science, chemical engineering, and biomedical engineering. Basic analysis of the wave structure inherent in the models as well as numerical discretization techniques are covered. Coding exercises/projects in combination with analytical calculations will be given on practical cases. The combination of such mathematical models, built on well-established physical laws and principles, and observation data provides a powerful approach to develop new computational tools for predicting unseen scenarios.

## Learning outcome

Knowledge:

The student will have an extended understanding of the following concepts:

1. Nonlinear and linear conservation laws.
2. Analytical solutions and numerical solutions.
3. Stability and accuracy of numerical approximations of the continuous model.
4. Know how to formulate a mathematical model to gain insight into phenomena related to fluid and traffic flow as illustrative examples.
5. Experience with how to write a scientific report.
6. Practical coding experience with Python through project work.
7. Experience with combining statistical methods like Monte Carlo Marko Chain (MCMC) and conservation law to identify unknown laws and structure behind observation data.

Skills:

The student will be able to

1. Understand how nonlinear and linear conservation laws naturally occur in the modeling of different dynamic processes that vary in space and time, man-made as well as phenomena found in nature.
2. Be able to construct analytical solutions of certain classes of problems.
3. Formulate discrete schemes that can be used to compute numerical solutions of general conservation laws.
4. Have an understanding of the challenges with finding solutions of nonlinear conservation laws and the need of concepts like shock wave and rarefaction wave, crossing characteristics, Rankine-Hugoniot jump condition, Riemann problem, entropy solution, and weak solutions.
5. Be able to implement in matlab/python a numerical scheme, compute and visualize approximate solutions.
6. Obtain insight into method that can be used to identify a conservation law behind a given set of observation data.

## Required prerequisite knowledge

Need to meet the requirement for admission to the master program in Computational Engineering, Petroleum Engineering and other equivalent master engineering programmes.

## Recommended prerequisites

Basic knowledge in mathematics (calculus) and physics. Some experience with coding (matlab/python) will be necessary. Basic knowledge of differential equations, numerical discretization and coding as presented in courses like MOD510/PET510 is recommended

## Coursework requirements

Mandatroy assignment (2-4)
Mandatory assignments (2-4) which must be approved 3 weeks before the exam.

Steinar Evje

Øystein Arild

## Method of work

Class room instruction, programming exercises, calculation exercises, project work.

## Open for

Admission to Single Courses at the Faculty of Science and Technology
Petroleum Engineering - Master of Science Degree Programme
Exchange programme at Faculty of Science and Technology

## Course assessment

There must be an early dialogue between the course supervisor, the student union representative and the students. The purpose is feedback from the students for changes and adjustments in the course for the current semester.In addition, a digital subject evaluation must be carried out at least every three years. Its purpose is to gather the students experiences with the course.

## Literature

Search for literature in Leganto