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Содержание 1 Physics-informed machine learning 1.1 Goals 1.2 Structure of the talk 1.3 Grading 1.4 Themes 1.5 Schedule

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]).

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

Grading

Each student presents two talks. Each talk lasts 25 minutes and concludes with a five-minute written test (discussion is better). 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. Highly recommended to present list of references 1 week before (for 1 point bonus reward, of course).

			First Theme					Second Theme
Student	Score	Seminars Work	T	WT	PT	QT	BT	T	WT	PT	QT	BT
Tikhonov Denis	8	0	Spherical harmonics for mechanical motion modelling	1	1	1	1	Modelling gravity with machine learning approaches	1	1	1	1
Panchenko Sviatoslav	8	1.5	Geometric Algebra, exterior product and quaternions	0	1	1	1	Conformal Geometric Algebra with Applications	0	1	1	1
Grigorev Alexey	8	0	Forward and Backward Fourier transform and iPhone LiDAR imaging analysis	1	1	1	1	Spectral Analysis on Meshes	1	1	1	1
Varenik Natalia	8	0	Continuous-Time Representation and Legendre Memory Units for BCI	1	1	1	1	Tensor representations of the Brain computer interfaces	1	1	1	1
Severilov Pavel	9	1	Fourier Neural Operator for Parametric Partial Differential Equations	1	1	1	1	Hamiltonian and Lagrangian Neural Networks	1	1	1	1
Petr Mokrov	9	0.5	Rectangular Flows	1	1	1	1	Dataset Dynamics	1	1	1	1
Kolesov Alexander	8	0	Geometric manifolds, Levy-Chivita operator and the curvature of tensors	1	1	1	1	SDE for generative modeling	1	1	1	1
Alexey Grishanov	3	0	High-order splines	0	1	1	1	-	0	0	0	0

T - theme; WT - text presented; PT - presentation presented; QT - questions presented; BT - bibliography presented.

All: text, questions and bibliography must be presented here. The presentation should be presented at the seminar and be in the table below.

Themes

1. Spherical harmonics for mechanical motion modelling
2. Geometric algebra, experior product and quaternions
3. Tensor representations of the Brain computer interfaces
4. Multi-view, kernels and metric spaces for the BCI and Brain Imaging
5. Continuous-Time Representation and Legendre Memory Units for BCI
6. Riemannian geometry on Shapes and diffeomorphisms for fMRI
7. The affine connection setting for transformation groups for fMRI
8. Strain, rotation and stress tensors modelling with examples
9. Differential forms and fibre bundles with examples
10. Modelling gravity with machine learning approaches
11. Geometric manifolds, the Levi-Chivita connection and curvature tensors
12. Flows and topological spaces
13. Application for Normalizing flow models (stress on spaces, not statistics)
14. Alignment in higher dimensions with RNN
15. Navier-Stokes equations and viscous flow
16. Newtonian and Non-Newtonian Fluids in Pipe Flows Using Neural Networks [1], [2]
17. Applications of Geometric Algebra and experior product
18. High-order splines
19. Forward and Backward Fourier transform and iPhone lidar imaging analysis
20. Fourier, cosine and Laplace transform for 2,3,4D and higher dimensions
21. Spectral analysis on meshes
22. 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
September 9	Spherical harmonics for mechanical motion modelling	Tikhonov Denis	slidesvideo
September 16	Geometric Algebra, exterior product and quaternions	Panchenko Sviatoslav	slides
October 7	Modelling gravity with machine learning approaches	Tikhonov Denis	slidesvideo
October 21	Forward and Backward Fourier transform and iPhone LiDAR imaging analysis	Grigorev Alexey	slidesvideo
October 21	Continuous-Time Representation and Legendre Memory Units for BCI	Varenik Natalia	slidesvideo
November 4	Fourier Neural Operator for Parametric Partial Differential Equations	Severilov Pavel	slidesvideo
November 4	Rectangular Flows	Petr Mokrov	slidesvideo
November 4	Spectral Analysis on Meshes	Grigorev Alexey	slidesvideo
November 4	Tensor representations of the Brain computer interfaces	Varenik Natalia	slidesvideo
November 18	Hamiltonian and Lagrangian Neural Networks	Severilov Pavel	slidesvideo
November 18	Dataset Dynamics	Petr Mokrov	slidesvideo
November 18	Geometric manifolds, Levy-Chivita operator and the curvature of tensors	Kolesov Alexander	slides
November 25	Conformal Geometric Algebra with Applications	Panchenko Sviatoslav	slides
November 25	SDE for generative modeling	Kolesov Alexander	slides
December 2	High-order splines	Grishanov Alexey	slides
December 2
December 9	Final discussion and grading	Andriy Graboviy