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Title: Identification of anomalous diffusion sources by unsupervised learning

Abstract

Fractional Brownian motion (fBm) is a ubiquitous diffusion process in which the memory effects of the stochastic transport result in the mean-squared particle displacement following a power law $$\langle$$Δr²$$\rangle$$ ~ tα, where the diffusion exponent α characterizes whether the transport is subdiffusive (α < 1), diffusive (α = 1), or superdiffusive (α > 1). Due to the abundance of fBm processes in nature, significant efforts have been devoted to the identification and characterization of fBm sources in various phenomena. In practice, the identification of the fBm sources often relies on solving a complex and ill-posed inverse problem based on limited observed data. In the general case, the detected signals are formed by an unknown number of release sources, located at different locations and with different strengths, that act simultaneously. This means that the observed data are composed of mixtures of releases from an unknown number of sources, which makes the traditional inverse modeling approaches unreliable. Here, we report an unsupervised learning method, based on non-negative matrix factorization, that enables the identification of the unknown number of release sources as well the anomalous diffusion characteristics based on limited observed data and the general form of the corresponding fBm Green’s function. We show that our method performs accurately for different types of sources and configurations with a predetermined number of sources with specific characteristics and introduced noise.

Authors:
ORCiD logo; ORCiD logo; ORCiD logo; ; ORCiD logo
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program
OSTI Identifier:
1631206
Alternate Identifier(s):
OSTI ID: 1648104
Report Number(s):
LA-UR-20-23646
Journal ID: ISSN 2643-1564; PPRHAI; 023248
Grant/Contract Number:  
20190020DR; 89233218CNA000001; 20190020DR,
Resource Type:
Published Article
Journal Name:
Physical Review Research
Additional Journal Information:
Journal Name: Physical Review Research Journal Volume: 2 Journal Issue: 2; Journal ID: ISSN 2643-1564
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Computer Science; Earth Sciences; Mathematics; Anomalous Diffusion; Unsupervised Learning

Citation Formats

Vangara, Raviteja, Rasmussen, Kim Ø., Petsev, Dimiter N., Bel, Golan, and Alexandrov, Boian S. Identification of anomalous diffusion sources by unsupervised learning. United States: N. p., 2020. Web. doi:10.1103/PhysRevResearch.2.023248.
Vangara, Raviteja, Rasmussen, Kim Ø., Petsev, Dimiter N., Bel, Golan, & Alexandrov, Boian S. Identification of anomalous diffusion sources by unsupervised learning. United States. doi:https://doi.org/10.1103/PhysRevResearch.2.023248
Vangara, Raviteja, Rasmussen, Kim Ø., Petsev, Dimiter N., Bel, Golan, and Alexandrov, Boian S. Fri . "Identification of anomalous diffusion sources by unsupervised learning". United States. doi:https://doi.org/10.1103/PhysRevResearch.2.023248.
@article{osti_1631206,
title = {Identification of anomalous diffusion sources by unsupervised learning},
author = {Vangara, Raviteja and Rasmussen, Kim Ø. and Petsev, Dimiter N. and Bel, Golan and Alexandrov, Boian S.},
abstractNote = {Fractional Brownian motion (fBm) is a ubiquitous diffusion process in which the memory effects of the stochastic transport result in the mean-squared particle displacement following a power law $\langle$Δr²$\rangle$ ~ tα, where the diffusion exponent α characterizes whether the transport is subdiffusive (α < 1), diffusive (α = 1), or superdiffusive (α > 1). Due to the abundance of fBm processes in nature, significant efforts have been devoted to the identification and characterization of fBm sources in various phenomena. In practice, the identification of the fBm sources often relies on solving a complex and ill-posed inverse problem based on limited observed data. In the general case, the detected signals are formed by an unknown number of release sources, located at different locations and with different strengths, that act simultaneously. This means that the observed data are composed of mixtures of releases from an unknown number of sources, which makes the traditional inverse modeling approaches unreliable. Here, we report an unsupervised learning method, based on non-negative matrix factorization, that enables the identification of the unknown number of release sources as well the anomalous diffusion characteristics based on limited observed data and the general form of the corresponding fBm Green’s function. We show that our method performs accurately for different types of sources and configurations with a predetermined number of sources with specific characteristics and introduced noise.},
doi = {10.1103/PhysRevResearch.2.023248},
journal = {Physical Review Research},
number = 2,
volume = 2,
place = {United States},
year = {2020},
month = {5}
}

Journal Article:
Free Publicly Available Full Text
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DOI: https://doi.org/10.1103/PhysRevResearch.2.023248

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Works referenced in this record:

Identification of contaminant sources in enclosed environments by inverse CFD modeling
journal, June 2007


Anomalous Transport in “Classical” Soil and Sand Columns
journal, January 2004

  • Cortis, Andrea; Berkowitz, Brian
  • Soil Science Society of America Journal, Vol. 68, Issue 5
  • DOI: 10.2136/sssaj2004.1539

Subdiffusion of nonlinear waves in quasiperiodic potentials
journal, October 2012


Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation
journal, October 2009


Signatures of mutational processes in human cancer
journal, August 2013

  • Alexandrov, Ludmil B.; Nik-Zainal, Serena; Wedge, David C.
  • Nature, Vol. 500, Issue 7463
  • DOI: 10.1038/nature12477

Silhouettes: A graphical aid to the interpretation and validation of cluster analysis
journal, November 1987


Superdiffusion dominates intracellular particle motion in the supercrowded cytoplasm of pathogenic Acanthamoeba castellanii
journal, June 2015

  • Reverey, Julia F.; Jeon, Jae-Hyung; Bao, Han
  • Scientific Reports, Vol. 5, Issue 1
  • DOI: 10.1038/srep11690

A coupled method for inverse source problem of spatial fractional anomalous diffusion equations
journal, October 2010


Application of inverse methods to contaminant source identification from aquitard diffusion profiles at Dover AFB, Delaware
journal, July 1999

  • Liu, Chongxuan; Ball, William P.
  • Water Resources Research, Vol. 35, Issue 7
  • DOI: 10.1029/1999WR900092

A tutorial on inverse problems for anomalous diffusion processes
journal, February 2015


The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics
journal, July 2004


Reconstruction of an order of derivative and a source term in a fractional diffusion equation from final measurements
journal, January 2018


Aging continuous time random walks
journal, April 2003

  • Barkai, Eli; Cheng, Yuan-Chung
  • The Journal of Chemical Physics, Vol. 118, Issue 14
  • DOI: 10.1063/1.1559676

Fractional Brownian motion and fractional Gaussian noise in subsurface hydrology: A review, presentation of fundamental properties, and extensions
journal, October 1997

  • Molz, F. J.; Liu, H. H.; Szulga, J.
  • Water Resources Research, Vol. 33, Issue 10
  • DOI: 10.1029/97WR01982

Parameter identification for mixed fractional Brownian motions with the drift parameter
journal, December 2019

  • Cai, Chunhao; Cheng, Xuwen; Xiao, Weilin
  • Physica A: Statistical Mechanics and its Applications, Vol. 536
  • DOI: 10.1016/j.physa.2019.04.178

Observation of Subdiffusion in a Disordered Interacting System
journal, June 2011


On principal component analysis, cosine and Euclidean measures in information retrieval
journal, November 2007


Estimating Diffusion Propagator and Its Moments Using Directional Radial Basis Functions
journal, October 2015

  • Ning, Lipeng; Westin, Carl-Fredrik; Rathi, Yogesh
  • IEEE Transactions on Medical Imaging, Vol. 34, Issue 10
  • DOI: 10.1109/TMI.2015.2418674

Maximum Likelihood from Incomplete Data Via the EM Algorithm
journal, September 1977

  • Dempster, A. P.; Laird, N. M.; Rubin, D. B.
  • Journal of the Royal Statistical Society: Series B (Methodological), Vol. 39, Issue 1
  • DOI: 10.1111/j.2517-6161.1977.tb01600.x

The random walk's guide to anomalous diffusion: a fractional dynamics approach
journal, December 2000


Exact maximum likelihood estimator for drift fractional Brownian motion at discrete observation
journal, September 2011


Weak Ergodicity Breaking in the Continuous-Time Random Walk
journal, June 2005


Anticorrelations and Subdiffusion in Financial Systems
journal, June 2003


Deciphering Signatures of Mutational Processes Operative in Human Cancer
journal, January 2013


Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications
journal, November 1990


Unsupervised phase mapping of X-ray diffraction data by nonnegative matrix factorization integrated with custom clustering
journal, August 2018

  • Stanev, Valentin; Vesselinov, Velimir V.; Kusne, A. Gilad
  • npj Computational Materials, Vol. 4, Issue 1
  • DOI: 10.1038/s41524-018-0099-2

Enhanced Diffusion in Active Intracellular Transport
journal, December 2000


Distributed non-negative matrix factorization with determination of the number of latent features
journal, February 2020


Communication: Consistent picture of lateral subdiffusion in lipid bilayers: Molecular dynamics simulation and exact results
journal, October 2011

  • Kneller, Gerald R.; Baczynski, Krzysztof; Pasenkiewicz-Gierula, Marta
  • The Journal of Chemical Physics, Vol. 135, Issue 14
  • DOI: 10.1063/1.3651800

An inverse problem for a generalized fractional diffusion
journal, December 2014

  • Furati, Khaled M.; Iyiola, Olaniyi S.; Kirane, Mokhtar
  • Applied Mathematics and Computation, Vol. 249
  • DOI: 10.1016/j.amc.2014.10.046

Photon Trajectories in Incoherent Atomic Radiation Trapping as Lévy Flights
journal, September 2004

  • Pereira, Eduardo; Martinho, José M. G.; Berberan-Santos, Mário N.
  • Physical Review Letters, Vol. 93, Issue 12
  • DOI: 10.1103/PhysRevLett.93.120201

Blind source separation for groundwater pressure analysis based on nonnegative matrix factorization
journal, September 2014

  • Alexandrov, Boian S.; Vesselinov, Velimir V.
  • Water Resources Research, Vol. 50, Issue 9
  • DOI: 10.1002/2013WR015037

Learning the parts of objects by non-negative matrix factorization
journal, October 1999

  • Lee, Daniel D.; Seung, H. Sebastian
  • Nature, Vol. 401, Issue 6755
  • DOI: 10.1038/44565

Hurst exponents, Markov processes, and fractional Brownian motion
journal, June 2007

  • McCauley, Joseph L.; Gunaratne, Gemunu H.; Bassler, Kevin E.
  • Physica A: Statistical Mechanics and its Applications, Vol. 379, Issue 1
  • DOI: 10.1016/j.physa.2006.12.028

Bayesian approach to a nonlinear inverse problem for a time-space fractional diffusion equation
journal, October 2018


Fractional Brownian Motions, Fractional Noises and Applications
journal, October 1968

  • Mandelbrot, Benoit B.; Van Ness, John W.
  • SIAM Review, Vol. 10, Issue 4
  • DOI: 10.1137/1010093

A fractal-based stochastic interpolation scheme in subsurface hydrology
journal, November 1993

  • Molz, Fred J.; Boman, Gerald K.
  • Water Resources Research, Vol. 29, Issue 11
  • DOI: 10.1029/93WR01914

Identification of release sources in advection–diffusion system by machine learning combined with Green’s function inverse method
journal, August 2018