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In recent work it has been established that deep neural networks are capable of approximating solutions to a large class of parabolic partial differential equations without incurring the curse of dimension.
Deep neural network approximation theory
D. Elbrächter, D. Perekrestenko, P. Grohs, and H. Bölcskei · 1901
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A theoretical analysis of deep neural networks and parametric pdes
G. Kutyniok, P. Petersen, M. Raslan, and R. Schneider · 1904
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M. Geist, P. Petersen, M. Raslan, R. Schneider, and G. Kutyniok · 2004
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T. von Petersdorff and C. Schwab · 2004
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F. Gazzola, H.-C. Grunau, and G. Sweers · 2010
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P. Mörters and Y. Peres · 2010
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Global Lipschitz regularity for a class of quasilinear elliptic equations
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On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations
W. E, M. Hutzenthaler, A. Jentzen, and T. Kruse · 2019
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Uniform error estimates for artificial neural network approximations for heat equations
L. Gonon, P. Grohs, A. Jentzen, D. Kofler, and D. Šiška · 2019
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Deep neural network approximations for Monte Carlo algorithms
P. Grohs, A. Jentzen, and D. Salimova · 2019
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Exponential relu dnn expression of holomorphic maps in high dimension
J. A. A. Opschoor, C. Schwab, and J. Zech · 2019
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Deep learning in high dimension: neural network expression rates for generalized polynomial chaos expansions in UQ
C. Schwab and J. Zech · 2019
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Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations
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