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Solving ODE/PDE with Neural Networks

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Differential equations and neural networks are naturally bonded. The best paper “Neural Ordinary Differential Equations” in NeurIPS 2018 caused a lot of attentions by utilizing ODE mechanisms when updating layer weights. On the other direction, there are also many research using neural network approaches to help investigate differential equations such as “Deep learning for universal linear embeddings of nonlinear dynamics”, “DGM: A deep learning algorithm for solving partial differential equations” or “Solving Irregular and Data-enriched Differential Equations using Deep Neural Networks”. In this post, I will make two toy examples to show the very the basic idea of using deep learning method for solving differential equations.

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Some intro

Recently I attended a workshop helping solve industrial problem hosted by the Fields Institute. One of the problems presented is developing accurate/efficient methods for matching Raman spectra from test sample to samples recorded in the library so that different chemicals can be detected effectively. This is a quite chanllenging problem. Theoretically speaking, though Raman spectroscopy is commonly used in chemistry to provide a structural fingerprint by which molecules can be identified, there are a huge amount of chemicals out in the nature among which many have quite similar Raman spectra. The fact that test samples are usually a mixture of different molecules make the problem even more difficult. Pratically speaking, spectra data recorded is not perfect. There will be noises of different kinds and background/baseline signal flooding the useful information.

Uottawa VPN connection via wifi

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It seems that the bash script provided by the university does not work for my machine with Ubuntu 16.04 LTS. For a wireless connection through VPN to be able to be “on campus”, you can follow the easy steps listed below.

• Run the following commends in the location where you saved the configuration file:
pcf2vpnc empl.pcf  empl.conf
sudo cp empl.conf /etc/vpnc/
sudo vpnc empl


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The Problem

I have a dual system window10/Ubuntu16.04 installed in my laptop. Today I can not access window files from Ubuntu and tried one command line from youtube which seems to mess things up :< The system did not boot like before but entering into the grub prompt instead. It seems that system does not know where/what to boot now and may need a manual configuration.

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Why Leaves?

From long time ago, people have already learned to identify different kinds of plants by examing their leaves. Nowadays, leaf Morphology, Taxonomy and Geometric Morphometrics are still actively investigated. Leaves are beautiful creations of nature, people today are frequently inspired by them for creations of art works. For example, Candian people use a maple leaf as the center of their flag. It would very nice if computers can help create leaves automatically from sratches. I assume this is a very difficult task. In order to make a beginner’s start, it may be beneficial to investigate what makes different leaves different from each other. This is a classification problem. Features learned from classification may help us have a peek at a glimpse of nature’s genius idea when it decides to make such creations. In industry, automatic recognition of plants is also useful for tasks such as species identification/reservation, automatic separate management in botany gardens or farms uses plants to produce medicines. It is also a good practice for me to learn things that are beyong textbooks.

geodesics on 2d surfaces

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Geometries can be fun, though it will require a deep and strict mathematical formulation. Physics and geometry has nature and deep connections with each other. It is well known that einstein’s theory of general relaltivity will need the support from non-Euclidean Geometry. One of the interesting topics in geometry is about finding geodesics. You can image the geodesic as a curve that minimizes certain metrics bewteen two fixed points in the considered space. One natural way of formulating the problem formally is by calculus of variations where you write your target metric as a functional and the Euler - Langrange Equation of this functional will govern the behavior of the geodesic. Usually, if the configuration space will have dimension 2N where N is the dimension of the manifold with geodesics on. There is a beatiful theorm: Noether’s theorem that can help deduct the dimension to N by solving a corresponding equation on the Lie Algebra. I think Prof. Terrence Tao’s post gives a very good explanation about this techique.

Blender Scenes

A collection of scenes I made with Blender. Please feel free to view more by clicking the title.

Some geodesics

Evolving landmarks with Thin-plate-spline deformation and an EP particle system

Deforming between shapes

Sunsets

Sunsets collections

A Conservative Formulation and a Numerical Algorithm for the Double-Gyre Nonlinear Shallow-Water Model

Published in Numerical Mathematics: Theory, Methods and Applications, 2015

This paper develops an algorithm for simulating a nonlinear shallow-water model.

Solitary Waves and N-Particle Algorithms for a Class of Euler–Poincaré Equations

Published in Studies in Applied Mathematics, 2016

This paper tries to extend Camassa-Holm equation in 2D case with a particle formulation. It received Highlights of the Year 2016 from the journal.

A Geodesic Landmark Shooting Algorithm for Template Matching and Its Applications

Published in SIAM J. Imaging Sci., 2016

This paper is about using the particle methods for feature extraction and some examples of utilizing it in the downstream cluster/classification task for shape outlines.

Characterization of Powder River Basin coal pyrolysis with cost-effective and environmentally friendly composite Na-Fe catalysts in a thermogravimetric analyzer and a fixed-bed reactor.

Published in International Journal of Hydrogen Energy, 2017

Recommended citation: Bang Xu, Dongyang Kuang, Fangjing Liu, Wenyang Lu, Alexander K.Goroncyc, Ting He, Khaled Gasem and Maohong Fan. Characterization of Powder River Basin coal pyrolysis with cost-effective and environmentally friendly composite Na-Fe catalysts in a thermogravimetric analyzer and a fixed-bed reactor International Journal of Hydrogen Energy.Volume 43, Issue 14, pp 6918-6935

A Landmark-based algorithm for automatic pattern recognition and abnormality detection.

Published in Pattern Recognition, 2018

This paper develops algorithms for constructing an “average” shape and some downstream applications based on landmark representations.

Predicting kinetic triplets using a 1d convolutional neural network.

Published in Thermochimica Acta, 2018

This paper develops an Covolutional Neural Network for the prediction of kinetic triplet.

Some Optional Methods of Activation Energy Determination on Pyrolysis

Published in Kinetics and Catalysis, 2019

Check the publication here

Cycle-consistent training for Reducing Negative Jacobian Determinant in Deep Registration Networks

Published in Lecture Notes in Computer Sciences, Springer, 2019

This paper apply a cycle consistent training for reducing negative jacobian determinant when deep networks are used for unsupervised registration tasks.

FAIM – A ConvNet Method for Unsupervised 3D Medical Image Registration

Published in Lecture Notes in Computer Sciences, Springer, 2019

This paper develops a Covolutional Neural Network for 3D Medical Image Registration.

Kinetics and mechanism of $CO_2$ gasification of coal catalyzed by $Na_2CO_3$, $FeCO_3$ and $Na_2CO_3-FeCO_3$

Published in Journal of the Energy Institute, 2019

Check the publication here

A Bayesian method on landmark momentum data for abnormality detection

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Using posterier predictive distribution to find outliers in the data.

2B or not 2B

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This talk is about classifying different jet plane’s countour using extracted momenta features using a particle formulation.

Medical image registration with neural networks

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This talk is about using neural networks for the task of medical image registration/alignment.

Spectra Matching

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Some ideas for improving current Raman spectra matching algorithms presented in a industrial problems solving workshop held by NRC-CNRC and Fields Institute

Covnets_in_image_registration

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Convnets, a different view of approximating diffeomorphisms in medical image registration. Video

Teaching @ UWYO

Undergraduate course, University of Wyoming, Math Department, 2012

From 2012 to 2016: Trigonometry & Algebra, Trigonometry, Calculus I

Teaching @ SUU, Fall 2016

Undergraduate course, Southern Utah University, Math Department, 2016

Intermediate Algebra, Trigonometry

Teaching @ SUU, Spring 2017

Undergraduate course, Southern Utah University, Department of Mathematics, 2017

Trigonometry and Algebra, Calculus I

Teaching @ UOttawa, 2017 - 2018

Undergraduate course, University of Ottawa, Department of Math & Stats, 2017

Calculus I and Mathematical Method I

Teaching @ UOttawa, 2018 - 2019

Undergraduate course, University of Ottawa, Department of Math & Stats, 2018

• Calculus I for Life Sciences
• Calculus II