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Cited by 57

The DOI matches its Crossref record; 72 of the 72 references it cites were found in a registry or an index.

Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks

Geist, Moritz, Petersen, Philipp, Raslan, Mones, Schneider, Reinhold, Kutyniok, Gitta·2021·Journal of Scientific Computing
DOI 10.1007/s10915-021-01532-w
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Abstract

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 model should depend only very mildly on the dimension of the parameter space and is determined by the intrinsic dimension of the solution manifold of the parametric partial differential equation. We use various methods to establish comparability between test-cases by minimizing the effect of the choice of test-cases on the optimization and sampling aspects of the learning problem. We find strong support for the hypothesis that approximation-theoretical effects heavily influence the practical behavior of learning problems in numerical analysis. Turning to practically more successful and modern architectures, at the end of this study we derive improved error bounds by focusing on convolutional neural networks.

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Crossref · 10.1007/s10915-021-01532-w

Cites

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Figures

Fig. 1 Partition of Omega1 as in test-case [T2] (left) for p = 9 , (the red and blue areas indicate the Omega1 i ), test-case [T3-F] (middle) for p = 4 (the red areas indicate the Omega1 i ) and test-case [T3-V] (right) for p = 8 (the red areas indicate the Omega1 i , y i + s 2 ) (Color figure online)
Fig. 1 Partition of Omega1 as in test-case [T2] (left) for p = 9 , (the red and blue areas indicate the Omega1 i ), test-case [T3-F] (middle) for p = 4 (the red areas indicate the Omega1 i ) and test-case [T3-V] (right) for p = 8 (the red areas indicate the Omega1 i , y i + s 2 ) (Color figure online)CC-BY · From the PDF
Fig. 2 Log-plot of the 200 largest normalized singular values for [T1] (top left), [T2] (top right), [T3]-V (bottom left) and [T4] (bottom right)
Fig. 2 Log-plot of the 200 largest normalized singular values for [T1] (top left), [T2] (top right), [T3]-V (bottom left) and [T4] (bottom right)CC-BY · From the PDF
Image 3
CC-BY · From the PDF
Image 4
CC-BY · From the PDF
Fig. 5 Plot of the mean relative test error for the sets of test-case [T1] for different values σ . The horizontal axis follows the dimension of the parameter space p the mean relative test error is shown on the vertical axis. Both axes use a logarithmic scale
Fig. 5 Plot of the mean relative test error for the sets of test-case [T1] for different values σ . The horizontal axis follows the dimension of the parameter space p the mean relative test error is shown on the vertical axis. Both axes use a logarithmic scaleCC-BY · From the PDF
Fig. 6 Plot of the mean relative test error for the sets of test-case [T2] for different values of μ with p on the horizontal axis and the mean relative test error on the vertical axis. Both axes use a logarithmic scale
Fig. 6 Plot of the mean relative test error for the sets of test-case [T2] for different values of μ with p on the horizontal axis and the mean relative test error on the vertical axis. Both axes use a logarithmic scaleCC-BY · From the PDF
Fig. 7 Plot of the mean relative test error for the sets of test-case [T3] with p on the horizontal axis and the error on the vertical axis. Both axes use a logarithmic scale
Fig. 7 Plot of the mean relative test error for the sets of test-case [T3] with p on the horizontal axis and the error on the vertical axis. Both axes use a logarithmic scaleCC-BY · From the PDF
Fig. 8 Plot of the mean relative test error for the sets of test-case [T4] with p on the horizontal axis and the error on the vertical axis. Only the horizontal axis is scaled logarithmically
Fig. 8 Plot of the mean relative test error for the sets of test-case [T4] with p on the horizontal axis and the error on the vertical axis. Only the horizontal axis is scaled logarithmicallyCC-BY · From the PDF
Fig. 9 CNN architecture used for experiments. Below each convolutional layer the number of filters is listed. The shrinking height and width of the blocks symbolize the lowered resolution after the red pooling layers. The visualization is created using the tools of www.github.com/HarisIqbal88/PlotNeuralNet
Fig. 9 CNN architecture used for experiments. Below each convolutional layer the number of filters is listed. The shrinking height and width of the blocks symbolize the lowered resolution after the red pooling layers. The visualization is created using the tools of www.github.com/HarisIqbal88/PlotNeuralNetCC-BY · From the PDF
Fig. 10 Plot of the mean relative test errors achieved with our CNN architecture for the sets of test-case [T2] (left), [T3-V] (middle) and [T4] (right). All axes except the vertical axis for [T4] are scaled logarithmically
Fig. 10 Plot of the mean relative test errors achieved with our CNN architecture for the sets of test-case [T2] (left), [T3-V] (middle) and [T4] (right). All axes except the vertical axis for [T4] are scaled logarithmicallyCC-BY · From the PDF
Fig. 11 Plot of the range of the mean relative test error over three runs for the sets of test-case [T1] . Both axes use a logarithmic scale
Fig. 11 Plot of the range of the mean relative test error over three runs for the sets of test-case [T1] . Both axes use a logarithmic scaleCC-BY · From the PDF
Fig. 12 Plot of the range of the mean relative test error over three runs for the sets of test-case [T2] . Both axes use a logarithmic scale
Fig. 12 Plot of the range of the mean relative test error over three runs for the sets of test-case [T2] . Both axes use a logarithmic scaleCC-BY · From the PDF
Fig. 13 Plot of the range of the mean relative test error over three runs for the sets of test-case [T3] . Both axes use a logarithmic scale
Fig. 13 Plot of the range of the mean relative test error over three runs for the sets of test-case [T3] . Both axes use a logarithmic scaleCC-BY · From the PDF
Fig. 14 Plot of the range of the mean relative test error over three runs for the sets of test-case [T4] . Only the horizontal axis is scaled logarithmically
Fig. 14 Plot of the range of the mean relative test error over three runs for the sets of test-case [T4] . Only the horizontal axis is scaled logarithmicallyCC-BY · From the PDF
Fig. 15 Plot of the mean relative training error for [T2] with p = 25 and different shifts μ
Fig. 15 Plot of the mean relative training error for [T2] with p = 25 and different shifts μCC-BY · From the PDF
Fig. 16 Plot of the mean relative training error for [T3-V] with p = 50 and different shifts μ
Fig. 16 Plot of the mean relative training error for [T3-V] with p = 50 and different shifts μCC-BY · From the PDF
Fig. 17 Plot of the mean relative training and test error for [T3-V] with p = 50 and μ = 10 -4
Fig. 17 Plot of the mean relative training and test error for [T3-V] with p = 50 and μ = 10 -4CC-BY · From the PDF
Fig. 18 Plot of the mean relative training and test error for [T4] with p = 91 and μ = 10 -1
Fig. 18 Plot of the mean relative training and test error for [T4] with p = 91 and μ = 10 -1CC-BY · From the PDF

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Citations

Cites

Research works printed but not linked: 8

  1. Adcock, B., Brugiapaglia, S., Dexter, N., Moraga, S.: Deep neural networks are effective at learning high-dimensional Hilbert-valued functions from limited data. arXiv preprint arXiv:2012.06081 (2020) · arXiv
  2. Berner, J., Grohs, P., Jentzen, A.: Analysis of the generalization error: empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations. arXiv preprint arXiv:1809.03062 (2018) · arXiv
  3. Bhattacharya, K., Hosseini, B., Kovachki, N.B., Stuart, A.M.: Model reduction and neural networks for parametric PDEs. arXiv preprintarXiv:2005.03180 (2020)
  4. Brevis, I., Muga, I., van der Zee, K.G.: Data-driven finite elements methods: machine learning acceleration of goal-oriented computations. arXiv preprint arXiv:2003.04485 (2020) · arXiv
  5. Jentzen, A., Salimova, D., Welti, T.: A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients. arXiv preprintarXiv:1809.07321 (2018)
  6. Khoo, Y., Lu, J., Ying, L.: Solving parametric PDE problems with artificial neural networks. arXiv preprint arXiv:1707.03351 (2017) · arXiv
  7. Marcati, C., Opschoor, J.A., Petersen, P.C., Schwab, C.: Exponential relu neural network approximation rates for point and edge singularities. arXiv preprint arXiv:2010.12217 (2020) · arXiv
  8. Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics informed deep learning (part ii): data-driven discovery of nonlinear partial differential equations. arxiv. arXiv preprint arXiv:1711.10561, (2017) · arXiv

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Metadata

Authors
Geist, M., Petersen, P., Raslan, M., Schneider, R., & Kutyniok, G.
Venue
Journal of Scientific Computing
Publisher
Springer Science and Business Media LLC
Published
2021-06-05
Volume
88
Issue
1
ISSN
  • 0885-7474
  • 1573-7691
DOI
10.1007/s10915-021-01532-w
Cited by
57
References
72
Language
en
Licence
Funders
University of Vienna

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