Modelling Physics with DAE

Master level course. Ecole Polytechnique 2022/23 (INF668 Cyber-Physical Systems)


Differential Algebraic Equations (DAE) arise as a natural modelling language for physical systems as they combine algebraic constraints and differential equations resulting from the laws of physics. This mathematical paradigm is at the basis of modelling languages like Modelica, a component-based language designed for the modeling of multi-physics systems. During this course we will be studying DAE from the computer scientist point of view, we will present the structural analysis at the heart of the state-of-the-art index reduction techniques. We will also present the main numerical integration schemes for DAE. The end of the course will be devoted to present the challenges reminiscent of supporting multi-mode DAE systems including the interactions with the nonsmooth behavior of some physical phenomena (e.g., impact laws), the multiple modes of operations, and the intrinsically discrete behavior of software components.

Keywords Modelling Physics, Differential Algebraic Equations, Structural Analysis, Numerical Integration, Compilation, Simulation


Graph Algorithms, Numerical Integration, Calculus, Abstract Algebra

Expected Outcome

The course aims at presenting the advantages, inherent difficulties, and future challenges of DAE-based programming languages like Modelica.

Roadmap (tentative)

  • January 26th. Lecture 1: General Introduction [Slides]
  • February 2nd. Lecture 2: Structural Analysis: Index Reduction [Slides]
  • February 7th. Lecture 3: Numerical Integration of DAEs [Slides]
  • February 16th. Lecture 5: Brief Introduction to Modelica [Online Tutorial]
  • March 9th. Lecture 4: Algebraic Methods [Slides]
  • March 14th. Lecture 6: Multi-mode DAE


The final score will be composed of two marks for the same project. Project proposals by December 20th.

  1. An oral presentation. March 23. 15-20mn Talk + 10-15mn Questions
  2. A Paper by March 20 Latex Template
    • Expected Sections: Introduction, Model, Simulation, Results, Related Work, Conclusion
    • To be sent by email.

Projects Topics (previous years)

  • External ballistics (trajectory of a projectile with a draft)
  • Spacetime curvature (Geodesic of dimension 1)
  • Arterial pressure as DAE
  • Coupled Motor-Propeller Model of a Quadcopter
  • Modeling greenhouse’s temperature and humidity levels dynamics




Internships (4 to 6 months) on related topics are available.
If interested, let me know as soon as possible.

Further Readings

cf. slides.

Office Hours

  • Thursday 2–4 PM (Inria Saclay)
  • For any request/question, feel free to email me (see address below)