@article{HijaziFreitagLandwehr2023,
  author    = {Hijazi, Saddam and Freitag, Melina A. and Landwehr, Niels},
  title     = {POD-Galerkin reduced order models and physics-informed neural networks for solving inverse problems for the Navier-Stokes equations},
  series = {Advanced modeling and simulation in engineering sciences : AMSES},
  volume    = {10},
  journal   = {Advanced modeling and simulation in engineering sciences : AMSES},
  number    = {1},
  publisher = {SpringerOpen},
  address   = {Berlin},
  issn      = {2213-7467},
  doi       = {10.1186/s40323-023-00242-2},
  pages     = {38},
  year      = {2023},
  abstract  = {We present a Reduced Order Model (ROM) which exploits recent developments in Physics Informed Neural Networks (PINNs) for solving inverse problems for the Navier-Stokes equations (NSE). In the proposed approach, the presence of simulated data for the fluid dynamics fields is assumed. A POD-Galerkin ROM is then constructed by applying POD on the snapshots matrices of the fluid fields and performing a Galerkin projection of the NSE (or the modified equations in case of turbulence modeling) onto the POD reduced basis. A POD-Galerkin PINN ROM is then derived by introducing deep neural networks which approximate the reduced outputs with the input being time and/or parameters of the model. The neural networks incorporate the physical equations (the POD-Galerkin reduced equations) into their structure as part of the loss function. Using this approach, the reduced model is able to approximate unknown parameters such as physical constants or the boundary conditions. A demonstration of the applicability of the proposed ROM is illustrated by three cases which are the steady flow around a backward step, the flow around a circular cylinder and the unsteady turbulent flow around a surface mounted cubic obstacle.},
  language  = {en}
}
@article{ShlapunovTarchanov2021,
  author    = {Shlapunov, Alexander and Tarchanov, Nikolaj Nikolaevič},
  title     = {An open mapping theorem for the Navier-Stokes type equations associated with the de Rham complex over R-n},
  series = {Siberian electronic mathematical reports = Sibirskie ėlektronnye matematičeskie izvestija},
  volume    = {18},
  journal   = {Siberian electronic mathematical reports = Sibirskie ėlektronnye matematičeskie izvestija},
  number    = {2},
  publisher = {Institut Matematiki Imeni S. L. Soboleva},
  address   = {Novosibirsk},
  issn      = {1813-3304},
  doi       = {10.33048/semi.2021.18.108},
  pages     = {1433 -- 1466},
  year      = {2021},
  abstract  = {We consider an initial problem for the Navier-Stokes type equations associated with the de Rham complex over R-n x[0, T], n >= 3, with a positive time T. We prove that the problem induces an open injective mappings on the scales of specially constructed function spaces of Bochner-Sobolev type. In particular, the corresponding statement on the intersection of these classes gives an open mapping theorem for smooth solutions to the Navier-Stokes equations.},
  language  = {en}
}
@article{MeraStepanenkoTarkhanov2018,
  author    = {Mera, Azal and Stepanenko, Vitaly A. and Tarkhanov, Nikolai Nikolaevich},
  title     = {Successive approximation for the inhomogeneous burgers equation},
  series = {Journal of Siberian Federal University : Mathematics \& Physics},
  volume    = {11},
  journal   = {Journal of Siberian Federal University : Mathematics \& Physics},
  number    = {4},
  publisher = {Siberian Federal University},
  address   = {Krasnoyarsk},
  issn      = {1997-1397},
  doi       = {10.17516/1997-1397-2018-11-4-519-531},
  pages     = {519 -- 531},
  year      = {2018},
  abstract  = {The inhomogeneous Burgers equation is a simple form of the Navier-Stokes equations. From the analytical point of view, the inhomogeneous form is poorly studied, the complete analytical solution depending closely on the form of the nonhomogeneous term.},
  language  = {en}
}
@article{MeraShlapunovTarkhanov2019,
  author    = {Mera, Azal and Shlapunov, Alexander A. and Tarkhanov, Nikolai Nikolaevich},
  title     = {Navier-Stokes Equations for Elliptic Complexes},
  series = {Journal of Siberian Federal University. Mathematics \& Physics},
  volume    = {12},
  journal   = {Journal of Siberian Federal University. Mathematics \& Physics},
  number    = {1},
  publisher = {Sibirskij Federalʹnyj Universitet},
  address   = {Krasnojarsk},
  issn      = {1997-1397},
  doi       = {10.17516/1997-1397-2019-12-1-3-27},
  pages     = {3 -- 27},
  year      = {2019},
  abstract  = {We continue our study of invariant forms of the classical equations of mathematical physics, such as the Maxwell equations or the Lam´e system, on manifold with boundary. To this end we interpret them in terms of the de Rham complex at a certain step. On using the structure of the complex we get an insight to predict a degeneracy deeply encoded in the equations. In the present paper we develop an invariant approach to the classical Navier-Stokes equations.},
  language  = {en}
}