Материал из MachineLearning.

Основная статья: Численные методы обучения по прецедентам (практика, В.В. Стрижов)

Содержание 1 Machine Learning for Theoretical Physics 1.1 Goals 1.2 Structure of the talk 1.3 Grading 1.4 Test 1.5 Themes 1.6 Schedule 1.7 General principles 1.8 Syllabus and goals 1.9 Theme 1: 1.9.1 Message 1.9.2 Basics 1.9.3 Application 1.9.4 Code 1.10 Theme 1: Manifolds 1.11 Code 1.12 Theme 1: ODE and flows 1.13 Theme 1: PDE 1.14 Theme 1: Navier-Stokes equations and viscous flow 1.15 Fourier for fun and practice 1D 1.16 Fourier for fun and practice nD 1.17 Geometric Algebra 1.18 Theme 1: High order splines 1.19 Theme 1: Topological data analysis 1.20 Theme 1: Homology versus homotopy 2 Fundamental theorems

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

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

Link to the template of the two-page essay.

Grading

Each student presents two talks. Each talk lasts 25 minutes and concludes with a five-minute written test.

Test

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

Themes

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. Flows and topological spaces
11. Application for Normalizing flow models (stress on spaces, not statistics)
12. Alignment in higher dimensions with RNN
13. Navier-Stokes equations and viscous flow
14. Newtonian and Non-Newtonian Fluids in Pipe Flows Using Neural Networks [1], [2]
15. Applications of Geometric Algebra and experior product
16. High-order splines
17. Forward and Backward Fourier transform and iPhone lidar imaging analysis
18. Fourier, cosine and Laplace transform for 2,3,4D and higher dimensions
19. Spectral analysis on meshes
20. 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

• Variational Inference with Normalizing Flows (source paper)
• Flow-based deep generative models

(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

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

W: Homology

Fundamental theorems

W: Inverse function theorem and Jacobian