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@ARTICLE{Guidetti:456491,
      author       = {Guidetti, V. and Muia, F. and Welling, Yvette Maria and
                      Westphal, Alexander},
      title        = {d{NN}solve: an efficient {NN}-based {PDE} solver},
      reportid     = {PUBDB-2021-01487, arXiv:2103.08662. DESY-21-037},
      pages        = {1-25},
      year         = {2021},
      note         = {16 pages + 9 pages of appendices, 7 figures, LaTeX, code to
                      be released soon},
      abstract     = {Neural Networks (NNs) can be used to solve Ordinary and
                      Partial Differential Equations (ODEs and PDEs) by redefining
                      the question as an optimization problem. The objective
                      function to be optimized is the sum of the squares of the
                      PDE to be solved and of the initial/boundary conditions. A
                      feed forward NN is trained to minimise this loss function
                      evaluated on a set of collocation points sampled from the
                      domain where the problem is defined. A compact and smooth
                      solution, that only depends on the weights of the trained
                      NN, is then obtained. This approach is often referred to as
                      PINN, from Physics Informed Neural
                      $Network~\cite{raissi2017physics_1,$ $raissi2017physics_2}.$
                      Despite the success of the PINN approach in solving various
                      classes of PDEs, an implementation of this idea that is
                      capable of solving a large class of ODEs and PDEs with good
                      accuracy and without the need to finely tune the
                      hyperparameters of the network, is not available yet. In
                      this paper, we introduce a new implementation of this
                      concept - called dNNsolve - that makes use of dual Neural
                      Networks to solve ODEs/PDEs. These include: i) sine and
                      sigmoidal activation functions, that provide a more
                      efficient basis to capture both secular and periodic
                      patterns in the solutions; ii) a newly designed
                      architecture, that makes it easy for the the NN to
                      approximate the solution using the basis functions mentioned
                      above. We show that dNNsolve is capable of solving a broad
                      range of ODEs/PDEs in 1, 2 and 3 spacetime dimensions,
                      without the need of hyperparameter fine-tuning.},
      cin          = {T},
      cid          = {I:(DE-H253)T-20120731},
      pnm          = {611 - Fundamental Particles and Forces (POF4-611) /
                      STRINGFLATION - Inflation in String Theory - Connecting
                      Quantum Gravity with Observations (647995)},
      pid          = {G:(DE-HGF)POF4-611 / G:(EU-Grant)647995},
      experiment   = {EXP:(DE-MLZ)NOSPEC-20140101},
      typ          = {PUB:(DE-HGF)25},
      eprint       = {2103.08662},
      howpublished = {arXiv:2103.08662},
      archivePrefix = {arXiv},
      SLACcitation = {$\%\%CITATION$ = $arXiv:2103.08662;\%\%$},
      doi          = {10.3204/PUBDB-2021-01487},
      url          = {https://bib-pubdb1.desy.de/record/456491},
}