Участник:Strijov/Drafts
Материал из MachineLearning.
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# Zongyi Li et al., Fourier Neural Operator for Parametric Partial Differential Equations // ICLR, 2021, [https://arxiv.org/abs/2010.08895 ArXiv] | # Zongyi Li et al., Fourier Neural Operator for Parametric Partial Differential Equations // ICLR, 2021, [https://arxiv.org/abs/2010.08895 ArXiv] | ||
- | ===Category theory== | + | ===Category theory=== |
# Tai-Danae Bradley, What is Applied Category Theory?, 2018, [https://arxiv.org/pdf/1809.05923.pdf ArXiv], [https://www.math3ma.com/ demo]. | # Tai-Danae Bradley, What is Applied Category Theory?, 2018, [https://arxiv.org/pdf/1809.05923.pdf ArXiv], [https://www.math3ma.com/ demo]. |
Версия 17:49, 6 сентября 2021
Machine Learning for Theoretical Physics
Physics-informed machine learning
(seminars by Andriy Graboviy and Vadim Strijov)
Goals
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]).
- Field and goals of a method or a model
- An overview of the method
- Notable authors and references
- Rigorous description, the theoretical part
- Algorithm and link to the code
- Application with plots
Grading
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.
Test
Todo: how make a test creative, not automised? Here be the test format.
Themes
- Spherical harmonics for mechanical motion modelling
- Tensor representations of the Brain computer interfaces
- Multi-view, kernels and metric spaces for the BCI and Brain Imaging
- Continuous-Time Representation and Legendre Memory Units for BCI
- Riemannian geometry on Shapes and diffeomorphisms for fMRI
- The affine connection setting for transformation groups for fMRI
- Strain, rotation and stress tensors modelling with examples
- Differential forms and fibre bundles with examples
- Modelling gravity with machine learning approaches
- Geometric manifolds, the Levi-Chivita connection and curvature tensors
- Flows and topological spaces
- Application for Normalizing flow models (stress on spaces, not statistics)
- Alignment in higher dimensions with RNN
- Navier-Stokes equations and viscous flow
- Newtonian and Non-Newtonian Fluids in Pipe Flows Using Neural Networks [1], [2]
- Applications of Geometric Algebra and experior product
- High-order splines
- Forward and Backward Fourier transform and iPhone lidar imaging analysis
- Fourier, cosine and Laplace transform for 2,3,4D and higher dimensions
- Spectral analysis on meshes
- Graph convolution and continuous Laplace operators
Schedule
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 |
9 | |||
9 | |||
16 | |||
16 | |||
23 | |||
23 | |||
30 | |||
30 | |||
October 7 | |||
7 | |||
14 | |||
14 | |||
21 | |||
21 | |||
28 | |||
28 | |||
November 4 | |||
4 | |||
11 | |||
11 | |||
18 | |||
18 | |||
25 | |||
25 | |||
December 2 | |||
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:
Message
Basics
Application
Code
https://papers.nips.cc/paper/2018/file/69386f6bb1dfed68692a24c8686939b9-Paper.pdf
Theme 1: Manifolds
Code
Surface differential geometry Coursera code video for Image and Video Processing
Theme 1: ODE and flows
- Neural Ordinary Differential Equations (source paper and code)
- W: Flow-based generative model
- Flows at deepgenerativemodels.github.io
- Знакомство с Neural ODE на хабре
Goes to BME
(after RBF)
Theme 1: PDE
Theme 1: Navier-Stokes equations and viscous flow
Fourier for fun and practice 1D
Fourier for fun and practice nD
See:
- 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
Fundamental theorems
W: Inverse function theorem and Jacobian
BCI, Matrix and tensor approximation
- Коренев, Г.В. Тензорное исчисление, 2000, 240 с., lib.mipt.ru.
- 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.
- Tai-Danae Bradley, At the Interface of Algebra and Statistics, 2020, ArXiv.
- Oseledets, I.V. Tensor-Train Decomposition //SIAM Journal on Scientific Computing, 2011, 33(5): 2295–2317, DOI, RG, lecture, GitHub, Tutoiral.
- Wikipedia: SVD, Multilinear subspace learning, HOSVD.
BCI, Feature selection
- Мотренко А.П.
High order partial least squares
Graph neural networks
- 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.
- Zhou, J. et al. Graph neural networks: A review of methods and applications // AI Open, 2020, 1: 57-81, DOI, ArXiv.
- 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.
- Zhang, S. et al. Graph convolutional networks: a comprehensive review // Computational Social Networks, 2019, 6(11), DOI.
- Xie, Y. et al. Self-Supervised Learning of Graph Neural Networks: A Unified Review // ArXiv.
- Wikipedia: Laplacian matrix, Discrete Poisson's equation, Graph FT
- GNN papers collection
Higher order Fourier transform
- Zongyi Li et al., Fourier Neural Operator for Parametric Partial Differential Equations // ICLR, 2021, ArXiv