Identification of release sources in advection–diffusion system by machine learning combined with Green's function inverse method

Valentin G. Stanev, Filip L. Iliev, Scott Hansen, Velimir V. Vesselinov, Boian S. Alexandrov

Research output: Contribution to journalArticlepeer-review


The identification of sources of advection–diffusion transport is based usually on solving complex ill-posed inverse models against the available state-variable data records. However, if there are several sources with different locations and strengths, the data records represent mixtures rather than the separate influences of the original sources. Importantly, the number of these original release sources is typically unknown, which hinders reliability of the classical inverse-model analyses. To address this challenge, we present here a novel hybrid method for identification of the unknown number of release sources. Our hybrid method, called HNMF, couples unsupervised learning based on Non-negative Matrix Factorization (NMF) and inverse-analysis Green's functions method. HNMF synergistically performs decomposition of the recorded mixtures, finds the number of the unknown sources and uses the Green's function of advection–diffusion equation to identify their characteristics. In the paper, we introduce the method and demonstrate that it is capable of identifying the advection velocity and dispersivity of the medium as well as the unknown number, locations, and properties of various sets of synthetic release sources with different space and time dependencies, based only on the recorded data. HNMF can be applied directly to any problem controlled by a partial-differential parabolic equation where mixtures of an unknown number of sources are measured at multiple locations.

Original languageAmerican English
Pages (from-to)64-76
Number of pages13
JournalApplied Mathematical Modelling
StatePublished - 1 Aug 2018
Externally publishedYes


  • Advection–diffusion transport
  • Green functions
  • Inverse problem
  • Non-negative matrix factorization
  • Source identification

All Science Journal Classification (ASJC) codes

  • Modelling and Simulation
  • Applied Mathematics


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