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Machine Learning for Theoretical Physics

Physics-informed machine learning
(seminars by Andriy Graboviy and Vadim Strijov)


The course consists of a series of group discussions devoted to various aspects of data modelling in continuous spaces. It will reduce the gap between the models of theoretical physics and the noisy measurements, performed under complex experimental circumstances. To show the selected neural network is an adequate parametrisation of the modelled phenomenon, we use geometrical axiomatic approach. We discuss the role of manifolds, tensors and differential forms in the neural network-based model selection.

The basics for the course are the book Geometric Deep Learning: April 2021 by Michael Bronstein et al. and the paper Physics-informed machine learning // Nature: May 2021 by George Em Karniadakis et al.

Structure of the talk

The talk is based on two-page essay ([template]).

  1. Field and goals of a method or a model
  2. An overview of the method
  3. Notable authors and references
  4. Rigorous description, the theoretical part
  5. Algorithm and link to the code
  6. Application with plots


Each student presents two talks. Each talk lasts 25 minutes and concludes with a five-minute written test. A seminar presentation gives 1 point, a formatted seminar text gives 1 point, a test gives 1 point, a reasonable test response gives 0.1 point. Bonus 1 point for a great talk.


Todo: how make a test creative, not automised? Here be the test format.


  1. Spherical harmonics for mechanical motion modelling
  2. Tensor representations of the Brain computer interfaces
  3. Multi-view, kernels and metric spaces for the BCI and Brain Imaging
  4. Continuous-Time Representation and Legendre Memory Units for BCI
  5. Riemannian geometry on Shapes and diffeomorphisms for fMRI
  6. The affine connection setting for transformation groups for fMRI
  7. Strain, rotation and stress tensors modelling with examples
  8. Differential forms and fibre bundles with examples
  9. Modelling gravity with machine learning approaches
  10. Geometric manifolds, the Levi-Chivita connection and curvature tensors
  11. Flows and topological spaces
  12. Application for Normalizing flow models (stress on spaces, not statistics)
  13. Alignment in higher dimensions with RNN
  14. Navier-Stokes equations and viscous flow
  15. Newtonian and Non-Newtonian Fluids in Pipe Flows Using Neural Networks [1], [2]
  16. Applications of Geometric Algebra and experior product
  17. High-order splines
  18. Forward and Backward Fourier transform and iPhone lidar imaging analysis
  19. Fourier, cosine and Laplace transform for 2,3,4D and higher dimensions
  20. Spectral analysis on meshes
  21. Graph convolution and continuous Laplace operators


Thursdays on 12:30 at m1p.org/go_zoom

  • September 2 9 16 23 30
  • October 7 14 21 28
  • November 4 11 18 25 
  • December 2 9

Date Theme Speaker Links
September 2 Course introduction and motivation Vadim Strijov GDL paper, Physics-informed
October 7
November 4
December 2
9 Final discussion and grading Andriy Graboviy

  • Geometric deep learning
  • Functional data analysis
  • Applied mathematics for machine learning

General principles

1. The experiment and measurements defines axioms i

Syllabus and goals

Theme 1:






Theme 1: Manifolds


Surface differential geometry Coursera code video for Image and Video Processing

Theme 1: ODE and flows

Goes to BME

(after RBF)

Theme 1: PDE

Theme 1: Navier-Stokes equations and viscous flow

Fourier for fun and practice 1D

Fourier Code

Fourier for fun and practice nD


  • Fourier analysis on Manifolds 5G page 49
  • Spectral analysis on meshes

Geometric Algebra

experior product and quaternions

Theme 1: High order splines

Theme 1: Topological data analysis

Theme 1: Homology versus homotopy

W: Homology

Fundamental theorems

W: Inverse function theorem and Jacobian

BCI, Matrix and tensor approximation

  1. Коренев, Г.В. Тензорное исчисление, 2000, 240 с., lib.mipt.ru.
  2.  Roger Penrose, "Applications of negative dimensional tensors," in Combinatorial Mathematics and its Applications, Academic Press (1971). See Vladimir Turaev, Quantum invariants of knots and 3-manifolds (1994), De Gruyter, p. 71 for a brief commentary PDF.
  3. Tai-Danae Bradley, At the Interface of Algebra and Statistics, 2020, ArXiv.
  4. Oseledets, I.V. Tensor-Train Decomposition //SIAM Journal on Scientific Computing, 2011, 33(5): 2295–2317, DOI, RG, lecture, GitHub, Tutoiral.
  5. Wikipedia: SVD, Multilinear subspace learning, HOSVD.

BCI, Feature selection

  1. Мотренко А.П.

High order partial least squares

=Graph neural networks

  1. Gama, F. et al. Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networks // IEEE Signal Processing Magazine, 2020, 37(6), 128-138, DOI.
  2. Zhou, J. et al. Graph neural networks: A review of methods and applications // AI Open, 2020, 1: 57-81, DOI, ArXiv.
  3. Zonghan, W. et al. A Comprehensive Survey on Graph Neural Networks // IEEE Transactions on Neural Networks and Learning Systems, 2021, 32(1): 4-24, DOI, ArXiv.
  4. Zhang, S. et al. Graph convolutional networks: a comprehensive review // Computational Social Networks, 2019, 6(11), DOI.
  5. Xie, Y. et al. Self-Supervised Learning of Graph Neural Networks: A Unified Review // ArXiv.
  6. Wikipedia: Laplacian matrix, Discrete Poisson's equation, Graph FT
  7. GNN papers collection

Higher order Fourier transform

  1. Zongyi Li et al., Fourier Neural Operator for Parametric Partial Differential Equations // ICLR, 2021, ArXiv

=Category theory

  1. Tai-Danae Bradley, What is Applied Category Theory?, 2018, ArXiv, demo.
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