SciML curriculum

Ordinary Differential Equations

Formal definition of ODEs, geometrical view, charged particle in a magnetic field

Video lectures

https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/resources/lecture-1-the-geometrical-view-of-y-f-x-y/

Reading materials

Trefethen: Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations https://people.maths.ox.ac.uk/trefethen/1all.pdf Chapter 1, sections 1

Numerical methods for solving ODEs

Explicit Euler, Implicit Euler, Trapezoid Rule, RK4

Video lectures

https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/resources/lecture-2-eulers-numerical-method-for-y-f-x-y/

Reading materials

Trefethen: Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations https://people.maths.ox.ac.uk/trefethen/1all.pdf Chapter 1, sections 2,3

Exercises

TUM SciComp 1, Worksheets 7, 8. Particularly the charged particle one

Brief Introduction to Partial Differential Equations

Definition of partial derivative 2D stationary heat equation 2D diffusion equation

Video lectures

• MIT 18.02 introduction to partial derivatives
  • [1]

Reading materials

• Trefethen: The (Unfinished) PDE coffee table book: Description of the heat equation
  • [2]

• Trefethen: Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations
  • https://people.maths.ox.ac.uk/trefethen/3all.pdf
  • Chapter 3, section 1

Finite Differences

Reading materials

• Trefethen: Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations
  • [3]
  • Chapter 3, Section 2

Exercises

TUM SciComp1, Worksheet 9.

1. 1. Neural nets

Interested in convolutional neural nets

Physics-informed neural nets

[4] [5]

Video lectures

• Differentiable Physics for Deep Learning, Overview Talk by Nils Thuerey
  • [6]

• Partial Differential Equations (PDEs), Convolutions, and the Mathematics of Locality
  • [7]

• Mixing Differential Equations and Neural Networks for Physics-Informed Learning
  • https://book.sciml.ai/notes/15/
  • [8]

Lecture notes (almost a textbook)

The above two video lectures are from a grad-level course on SciML:

• • [9]
  • [10]

Papers

• Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• An improved data-free surrogate model for solving partial differential equations using deep neural networks
• NeuralPDE: Automating Physics-Informed Neural Networks (PINNs) with Error Approximations
• Neural Ordinary Differential Equations