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

• 16.90 Lecture notes
  • Lecture 1: Numerical Integration of Ordinary Differential Equations
  • Lecture 6: Runge-Kutta Methods

• 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

• Iserles, Numerical Analysis of Differential Equations
  • Chapters 1,2 (maybe some of chapter 3)

Exercises

• TUM SciComp 1, Worksheet 6. Exercises 3,4
• TUM SciComp 1, Worksheet 7. Exercise 1,2,3

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
  • https://ocw.mit.edu/courses/18-02-multivariable-calculus-fall-2007/resources/lecture-8-partial-derivatives/

Reading materials

• Trefethen: The (Unfinished) PDE coffee table book: Description of the heat equation
  • https://people.maths.ox.ac.uk/trefethen/pdectb/heat2.pdf

• 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

• TUM SciComp Lecture 5

Finite Differences

Reading materials

• TUM SciComp, Lecture 5

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

• Iserles, Numerical Analysis of Differential Equations
  • Chapter 16, The Diffusion Equation: Sections 1,3

Exercises

• TUM SciComp1, Worksheet 9.
• TUM SciComp Lab, Worksheet 4.

Neural nets

Interested in convolutional neural nets

Physics-informed neural nets

• https://en.wikipedia.org/wiki/Physics-informed_neural_networks
• https://github.com/tum-pbs/Physics-Based-Deep-Learning

Video lectures

• Differentiable Physics for Deep Learning, Overview Talk by Nils Thuerey
  • https://www.youtube.com/watch?v=BwuRTpTR2Rg

• Partial Differential Equations (PDEs), Convolutions, and the Mathematics of Locality
  • https://www.youtube.com/watch?v=apkyk8n0vBo

• Mixing Differential Equations and Neural Networks for Physics-Informed Learning
  • https://book.sciml.ai/notes/15/
  • https://www.youtube.com/watch?v=YuaVXt--gAA

Lecture notes (almost a textbook)

• The above two video lectures are from a grad-level course on SciML:
  • https://mitmath.github.io/18337/
  • https://book.sciml.ai/

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