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=Machine Learning for Theoretical Physics=
 
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Physics-informed machine learning<br>
 
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(seminars by Andriy Graboviy and Vadim Strijov)
 
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==Goals==
 
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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.
 
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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.
 
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==Structure of the talk==
 
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# Field and goals of a method or a model
 
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# An overview of the method
 
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# Notable authors and references
 
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# Rigorous description, the theoretical part
 
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# Algorithm and link to the code
 
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# Application with plots
 
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Link to the template of the two-page essay.
 
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==Grading==
 
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Each student presents two talks. Each talk lasts 25 minutes and concludes with a five-minute written test.
 
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==Test==
 
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Todo: how make a test creative, not automised? Here be the test format.
 
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==Themes==
 
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# Spherical harmonics for mechanical motion modelling
 
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# Tensor representations of the Brain computer interfaces
 
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# Multi-view, kernels and metric spaces for the BCI and Brain Imaging
 
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# Continuous-Time Representation and Legendre Memory Units for BCI
 
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# Riemannian geometry on Shapes and diffeomorphisms for fMRI
 
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# The affine connection setting for transformation groups for fMRI
 
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# Strain, rotation and stress tensors modelling with examples
 
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# Differential forms and fibre bundles with examples
 
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# Modelling gravity with machine learning approache
 
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# Flows and topological spaces
 
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# Application for Normalizing flow models (stress on spaces, not statistics)
 
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# Alignment in higher dimensions with RNN
 
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# Navier-Stokes equations and viscous flow
 
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# Newtonian and Non-Newtonian Fluids in Pipe Flows Using Neural Networks [https://doi.org/10.2202/1556-3758.1079], [https://doi.org/10.1103/PhysRevE.102.043309]
 
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# Applications of Geometric Algebra and experior product
 
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# High-order splines
 
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# Forward and Backward Fourier transform and iPhone lidar imaging analysis
 
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# Fourier, cosine and Laplace transform for 2,3,4D and higher dimensions
 
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# Spectral analysis on meshes
 
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# Graph convolution and continuous Laplace operators
 
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==Schedule==
 
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Thursdays on 12:30 at m1p.org/go_zoom
 
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* September 2 9 16 23 30
 
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* October 7 14 21 28
 
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* November 4 11 18 25 
 
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* December 2 9
 
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{|class="wikitable"
 
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! Date
 
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! Theme
 
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! Speaker
 
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! Links
 
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|September 2
 
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|Course introduction and motivation
 
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|Vadim Strijov
 
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|[https://geometricdeeplearning.com/ GDL paper], [https://www.researchgate.net/publication/351814752_Physics-informed_machine_learning/link/60ae8f43a6fdcc647ede90f7/download Physics-informed]
 
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|Final discussion and grading
 
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|Andriy Graboviy
 
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* Geometric deep learning
 
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* Functional data analysis
 
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* Applied mathematics for machine learning
 
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==General principles==
 
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1. The experiment and measurements defines axioms i
 
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==Syllabus and goals==
 
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==Theme 1: ==
 
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===Message===
 
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===Basics===
 
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===Application===
 
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===Code===
 
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https://papers.nips.cc/paper/2018/file/69386f6bb1dfed68692a24c8686939b9-Paper.pdf
 
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==Theme 1: Manifolds ==
 
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==Code==
 
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Surface differential geometry
 
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[https://www.coursera.org/lecture/image-processing/3-surface-differential-geometry-duration-11-43-vtuJ1 Coursera code video] for Image and Video Processing
 
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==Theme 1: ODE and flows==
 
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*[https://papers.nips.cc/paper/2018/hash/69386f6bb1dfed68692a24c8686939b9-Abstract.html Neural Ordinary Differential Equations] (source paper and code)
 
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*[https://en.wikipedia.org/wiki/Flow-based_generative_model W: Flow-based generative model]
 
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*[https://deepgenerativemodels.github.io/notes/flow/ Flows at deepgenerativemodels.github.io]
 
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*[https://habr.com/ru/company/ods/blog/442002/ Знакомство с Neural ODE на хабре]
 
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Goes to BME
 
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*[https://arxiv.org/pdf/1505.05770.pdf Variational Inference with Normalizing Flows (source paper)]
 
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*[https://lilianweng.github.io/lil-log/2018/10/13/flow-based-deep-generative-models.html Flow-based deep generative models]
 
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(after RBF)
 
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==Theme 1: PDE==
 
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==Theme 1: Navier-Stokes equations and viscous flow==
 
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== Fourier for fun and practice 1D==
 
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[https://morioh.com/p/18b3158eab36?f=5c21fb01c16e2556b555ab32&fbclid=IwAR0FBF6IfmEaaedMEDzxiSclxGgNVweHgDQZympUZ-4doeKGNwQwUvp9upo Fourier Code]
 
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== Fourier for fun and practice nD==
 
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See:
 
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* Fourier analysis on Manifolds 5G page 49
 
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* Spectral analysis on meshes
 
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== Geometric Algebra ==
 
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experior product and quaternions
 
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==Theme 1: High order splines==
 
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==Theme 1: Topological data analysis==
 
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==Theme 1: Homology versus homotopy==
 
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[https://en.wikipedia.org/wiki/Homology_(mathematics) W: Homology]
 

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Fundamental theorems

W: Inverse function theorem and Jacobian

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