Efficient approximation of solutions of parametric linear transport equations by ReLU DNNs
We demonstrate that deep neural networks with the ReLU activation function can efficiently approximate the solutions of various types of parametric linear transport equations. For non-smooth initial conditions, the solutions of these PDEs are high-dimensional and non-smooth. Therefore, approximation...
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| Published in | Advances in computational mathematics Vol. 47; no. 1 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.02.2021
Springer Nature B.V |
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| Online Access | Get full text |
| ISSN | 1019-7168 1572-9044 |
| DOI | 10.1007/s10444-020-09834-7 |
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| Abstract | We demonstrate that deep neural networks with the ReLU activation function can efficiently approximate the solutions of various types of parametric linear transport equations. For non-smooth initial conditions, the solutions of these PDEs are high-dimensional and non-smooth. Therefore, approximation of these functions suffers from a curse of dimension. We demonstrate that through their inherent compositionality deep neural networks can resolve the characteristic flow underlying the transport equations and thereby allow approximation rates independent of the parameter dimension. |
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| AbstractList | We demonstrate that deep neural networks with the ReLU activation function can efficiently approximate the solutions of various types of parametric linear transport equations. For non-smooth initial conditions, the solutions of these PDEs are high-dimensional and non-smooth. Therefore, approximation of these functions suffers from a curse of dimension. We demonstrate that through their inherent compositionality deep neural networks can resolve the characteristic flow underlying the transport equations and thereby allow approximation rates independent of the parameter dimension. |
| ArticleNumber | 11 |
| Author | Petersen, Philipp Laakmann, Fabian |
| Author_xml | – sequence: 1 givenname: Fabian surname: Laakmann fullname: Laakmann, Fabian organization: Mathematical Institute, University of Oxford – sequence: 2 givenname: Philipp orcidid: 0000-0003-3566-1020 surname: Petersen fullname: Petersen, Philipp email: Philipp.Petersen@univie.ac.at organization: Institut für Mathematik, Universität Wien |
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| Keywords | 41A46 Deep neural networks 35Q49 41A25 Transport equations 68T05 35A35 Parametric PDEs Approximation rates Curse of dimension 65N30 |
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Complex.2009254398404254203910.1016/j.jco.2008.11.002 GoodfellowIBengioYCourvilleADeep Learning2016CambridgeMIT Press1373.68009http://www.deeplearningbook.org Montanelli, H., Yang, H., Du, Q.: Deep ReLU networks overcome the curse of dimensionality for bandlimited functions. arXiv:1903.00735 (2019) Gühring, I., Kutyniok, G., Petersen, P.: Error bounds for approximations with deep ReLU neural networks in Ws,p norms. 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| References_xml | – reference: Evans, L.: Partial differential equations. American Mathematical Society (2010) – reference: Dahmen, W., Gruber, F., Mula, O.: An adaptive nested source term iteration for radiative transfer equations. Mathematics of Computation (2020) – reference: He, J., Li, L., Xu, J., Zheng, C.: ReLU deep Neural Networks and Linear Finite Elements. arXiv:1807.03973 (2018) – reference: SirignanoJSpiliopoulosKDGM: a deep learning algorithm for solving partial differential equationsJ. Comput. Phys.201837513391364387458510.1016/j.jcp.2018.08.029 – reference: EWYuBThe deep ritz method: a deep learning-based numerical algorithm for solving variational problemsCommunications in Mathematics and Statistics201861112376795810.1007/s40304-018-0127-z – reference: Allaire, G., Blanc, X., Després, B., Golse, F.: Transport et diffusion. Ecole Polytechnique (2019) – reference: GoodfellowIBengioYCourvilleADeep Learning2016CambridgeMIT Press1373.68009http://www.deeplearningbook.org – reference: DahmenWPleskenCWelperGDouble greedy algorithms: reduced basis methods for transport dominated problemsESAIM: Mathematical Modelling and Numerical Analysis2014483623663317786010.1051/m2an/2013103 – reference: DiPernaRJLionsPLOrdinary differential equations, transport theory and Sobolev spacesInvent. Math.1989983511547102230510.1007/BF01393835 – reference: Serre, D.: Systems of Conservation Laws 1. Cambridge University Press (1999) – reference: Hutzenthaler, M., Jentzen, A., Kruse, T., Nguyen, T.A.: A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations. arXiv:1901.10854(2019) – reference: Suzuki, T.: Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality. arXiv:1810.08033 (2018) – reference: BarronARUniversal approximation bounds for superpositions of a sigmoidal functionIEEE Transactions on Information Theory1993393930945123772010.1109/18.256500 – reference: Opschoor, J.A.A., Schwab, C., Zech, J.: Exponential ReLU DNN expression of holomorphic maps in high dimension. Technical Report 2019-35, Seminar for Applied Mathematics, ETH Zürich, Switzerland (2019) – reference: Beck, C., Becker, S., Grohs, P., Jaafari, N., Jentzen, A.: Solving stochastic differential equations and kolmogorov equations by means of deep learning. arXiv:1806.00421 (2018) – reference: EggerHSchlottbomMA mixed variational framework for the radiative transfer equationMathematical Models and Methods in Applied Sciences201222031150014289045210.1142/S021820251150014X – reference: Elbrächter, D., Grohs, P., Jentzen, A., Schwab, C.: DNN expression rate analysis of high-dimensional PDEs: application to option pricing. arXiv:1809.07669 (2018) – reference: JohnFPartial differential equations1978USSpringer10.1007/978-1-4684-0059-5 – reference: YarotskyDError bounds for approximations with deep reLU networksNeural Netw.20179410311410.1016/j.neunet.2017.07.002 – reference: BölcskeiHGrohsPKutyniokGPetersenPOptimal approximation with sparsely connected deep neural networksSIAM Journal on Mathematics of Data Science201911845394969910.1137/18M118709X – reference: Schmidt-Hieber, J.: Deep ReLU network approximation of functions on a manifold. arXiv:1908.00695 (2019) – reference: Montanelli, H., Yang, H., Du, Q.: Deep ReLU networks overcome the curse of dimensionality for bandlimited functions. arXiv:1903.00735 (2019) – reference: Kutyniok, G., Petersen, P., Raslan, M., Schneider, R.: A theoretical analysis of deep neural networks and parametric PDEs. arXiv:1904.00377(2019) – reference: PoggioTMhaskarHRosascoLMirandaBLiaoQWhy and when can deep-but not shallow-networks avoid the curse of dimensionality: a reviewInt. J. Autom. Comput.201714550351910.1007/s11633-017-1054-2 – reference: Ambrosio, L., Gigli, N., Savare, G.: Gradient Flows. Birkhäuser-Verlag (2005) – reference: SchwabCZechJDeep learning in high dimension: neural network expression rates for generalized polynomial chaos expansions in UQAnal. Appl.201917011955389473210.1142/S0219530518500203 – reference: Laakmann, F.: A theoretical analysis of high-dimensional parametric transport equations and neural networks. Project thesis, University of Oxford (2019) – reference: LeCunYBengioYHintonGDeep learningNature2015521755343644410.1038/nature14539 – reference: Grella, K.: Sparse tensor approximation for radiative transport. PhD thesis, ETH Zurich (2013) – reference: NovakEWoźniakowskiHApproximation of infinitely differentiable multivariate functions is intractableJ. Complex.2009254398404254203910.1016/j.jco.2008.11.002 – reference: Hartman, P.: Ordinary differential equations. 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| SubjectTerms | Approximation Artificial neural networks Computational mathematics Computational Mathematics and Numerical Analysis Computational Science and Engineering Initial conditions Mathematical analysis Mathematical and Computational Biology Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Neural networks Transport equations Visualization |
| Title | Efficient approximation of solutions of parametric linear transport equations by ReLU DNNs |
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