Blar i forfatter "Petersen, Philipp"

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    • Deep microlocal reconstruction for limited-angle tomography 

      Andrade-Loarca, Héctor; Kutyniok, Gitta Astrid Hildegard; Öktem, Ozan; Petersen, Philipp (Journal article; Tidsskriftartikkel; Peer reviewed, 2022-01-04)
      We present a deep-learning-based algorithm to jointly solve a reconstruction problem and a wavefront set extraction problem in tomographic imaging. The algorithm is based on a recently developed digital wavefront set extractor as well as the well-known microlocal canonical relation for the Radon transform. We use the wavefront set information about x-ray data to improve the reconstruction ...

    • Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks 

      Geist, Moritz; Petersen, Philipp; Raslan, Mones; Schneider, Reinhold; Kutyniok, Gitta Astrid Hildegard (Journal article; Tidsskriftartikkel; Peer reviewed, 2021-06-05)
      We perform a comprehensive numerical study of the effect of approximation-theoretical results for neural networks on practical learning problems in the context of numerical analysis. As the underlying model, we study the machine-learning-based solution of parametric partial differential equations. Here, approximation theory for fully-connected neural networks predicts that the performance of the ...

    • A Theoretical Analysis of Deep Neural Networks and Parametric PDEs 

      Kutyniok, Gitta Astrid Hildegard; Petersen, Philipp; Raslan, Mones; Schneider, Reinhold (Journal article; Tidsskriftartikkel; Peer reviewed, 2021-06-02)
      We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent low dimensionality of the solution manifold to obtain approximation rates which are significantly superior to those provided by classical neural network approximation results. ...