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# 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]).

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.

## Test

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

## 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

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

# Fundamental theorems

Mathematical methods of forecasting

The lecture course and seminar introduces and applies methods of modern physics to the problems of machine learning.

Minimum topics to discuss: Geometric deep learning approach.

Optimum topics to discuss are: tensors, differential forms, Riemannian and differential geometry, metrics, differential operators in various spaces, embeddings, manifolds, bundles. We investigate scalar, vector and tensor fields (as well as jets, fibers and shiefs, tensor bundles, sheaf bundles etc.). The fields and spaces are one-dimensional, multidimensional and continuously dimensional.

### 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. Выбор моделей прогнозирования мультикоррелирующих временных рядов (диссертация), 2019 PDF
2. Исаченко Р.В. Снижение размерности пространства в задачах декодирования сигналов (дисссертация), 2021 PDF

### High order partial least squares

1. Qibin Zhao, et al. and A. Cichocki, Higher Order Partial Least Squares (HOPLS): A Generalized Multilinear Regression Method // IEEE Transactions on Pattern Analysis and Machine Intelligence, July 2013, pp. 1660-1673, vol. 35, DOI, ArXiv.

### Neural ODEs and Continuous normalizing flows

1. Ricky T. Q. Chen et al., Neural Ordinary Differential Equations // NIPS, 2018, ArXiv, source paper and code
2. Johann Brehmera and Kyle Cranmera, Flows for simultaneous manifold learning and density estimation // NIPS, 2020, ArXiv
3. Flows at deepgenerativemodels.github.io
4. Flow-based deep generative models
5. Variational Inference with Normalizing Flows (source paper, Goes to BME)
6. Знакомство с Neural ODE на хабре, W: Flow-based generative model

### Continous time representation

1. Самохина Алина, Непрерывное представление времени в задачах декодирования сигналов (магистерская диссертация): 2021 PDF, GitHub
2. Aaron R Voelker, Ivana Kajić, Chris Eliasmith, Legendre Memory Units: Continuous-Time Representation in Recurrent Neural Networks // NIPS, 2019, PDF,PDF.
3. Functional data analysis: splines

### Metric tensors and kernels

1. Lynn Houthuys and Johan A. K. Suykens, Tensor Learning in Multi-view Kernel PCA // ICANN 2018, pp 205-215, DOI.

### fRMI, Riemannian geometry on shapes

1. Xavier Pennec, Stefan Sommer, and Tom Fletcher, Riemannian Geometric Statistics in Medical Image Analysis, 2019 book
2. Surface differential geometry Coursera code video for Image and Video Processing

### Spatial time series alignment

1. Titouan Vayer et al., Time Series Alignment with Global Invariances, 2020,ArXiv
2. Marco Cuturi and Mathieu Blondel, Soft-DTW: a Differentiable Loss Function for Time-Series, ArXiv
3. Marcel Campen et al., Scale-Invariant Directional Alignment of Surface Parametrizations // Eurographics Symposium on Geometry Processing, 2016, 35(5), DOI
4. Helmut Pottmann et al. Geodesic Patterns // ACM Transactions on Graphics, 29(4), DOI, PDF

### Reproducing kernel Hilbert space

1. Mauricio A. Alvarez, Lorenzo Rosasco, Neil D. Lawrence, Kernels for Vector-Valued Functions: a Review, 2012, ArXiv
2. Pedro Domingos, Every Model Learned by Gradient Descent Is Approximately a Kernel Machine, 2020, ArXiv
3. Wikipedia: RKHS
4. Aronszajn, Nachman (1950). "Theory of Reproducing Kernels". Transactions of the American Mathematical Society. 68 (3): 337–404. DOI.

### Convolutions and Graphs

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
2. Fourier for fun and practice 1D Fourier Code
3. Fourier for fun and practice nD
• Fourier analysis on Manifolds 5G page 49
• Spectral analysis on meshes

### Spherical Regression

1. Shuai Liao, Efstratios Gavves, Cees G. M. Snoek, Spherical Regression: Learning Viewpoints, Surface Normals and 3D Rotations on N-Spheres // CVPR, 2019, 9759-9767, ArXiv

### Category theory

1. Tai-Danae Bradley, What is Applied Category Theory?, 2018, ArXiv, demo.
2. F. William Lawvere, Conceptual Mathematics: A First Introduction to Categories, 2011, PDF.
3. Картан А. Дифференциальное исчисление. Дифференциальные формы, 1971 lib.mipt.ru
4. Wikipedia: Homology, Topological data analysis

### Geometric algebra

1. experior product and quaternions
2. Nick Lucid, Advanced Theoretical Physics, 2019, sample.