
Mathematics in Engineering, 2018, 1(1): 118146. doi: 10.3934/Mine.2018.1.118
Research article
Export file:
Format
 RIS(for EndNote,Reference Manager,ProCite)
 BibTex
 Text
Content
 Citation Only
 Citation and Abstract
A machine learning framework for data driven acceleration of computations of di erential equations
Seminar for Applied Mathematics (SAM), DMath, ETH Z¨urich, R¨amistrasse 101, Z¨urich8092, Switzerland
Received: , Accepted: , Published:
References
1. Barron AR (1993) Universal approximation bounds for superpositions of a sigmoidal function. IEEE T Inform Theory 39: 930–945.
2. Beck C, Weinan E and Jentzen A (2017) Machine learning approximation algorithms for high dimensional fully nonlinear partial differential equations and secondorder backward stochastic differential equations. Technical Report 201749, Seminar for Applied Mathematics, ETH Zürich.
3. Bijl H, Lucor D, Mishra S, et al. (2014) Uncertainty quantification in computational fluid dynamics. Lecture notes in computational science and engineering 92, Springer.
4. Borzi A and Schulz V (2012) Computational optimization of systems governed by partial differential equations, SIAM.
5. Brenner SC and Scott LR (2008) The mathematical theory of finite element methods. Texts in applied mathematics 15, Springer.
6. Dafermos CM (2005) Hyperbolic Conservation Laws in Continuum Physics (2nd Ed.), Springer Verlag.
7. Weinan E and Yu B (2018) The deep Ritz method: a deep learningbased numerical algorithm for solving variational problems. Commun Math Stat 6: 1–12.
8. Fjordholm US, Mishra S and Tadmor E (2012) Arbitrarily highorder order accurate essentially nonoscillatory entropy stable schemes for systems of conservation laws. SIAM J Numer Anal 50: 544–573.
9. Ghanem R, Higdon D and Owhadi H (2016) Handbook of uncertainty quantification, Springer.
10. Godlewski E and Raviart PA (1991) Hyperbolic Systems of Conservation Laws. Mathematiques et Applications, Ellipses Publ., Paris.
11. Goodfellow I, Bengio Y and Courville A (2016) Deep learning. MIT Press. Available from: http://www.deeplearningbook.org.
12. Hairer E and Wanner G (1991) Solving ordinary differential equations. Springer Series in computational mathematics, 14, Springer.
13. Hornik K, Stinchcombe M, and White H (1989) Multilayer feedforward networks are universal approximators. Neural networks 2: 359–366.
14. Kingma DP and Ba JL (2015) Adam: a Method for Stochastic Optimization. International Conference on Learning Representations, 1–13.
15. LeCun Y, Bengio Y and Hinton G (2015) Deep learning. Nature 521: 436–444.
16. LeVeque RJ (2007) Finite difference methods for ordinary and partial differential equations, steady state and time dependent problems, SIAM.
17. Mishra S, Schwab C and Šukys J (2012) Multilevel Monte Carlo finite volume methods for nonlinear systems of conservation laws in multidimensions. J Comput Phys 231: 3365–3388.
18. Miyanawala TP and Jaiman RK (2017) An effcient deep learning technique for the NavierStokes equations: application to unsteady wake flow dynamics. Preprint, arXiv :1710.09099v2.
19. Poon H and Domingos P (2011) Sumproduct Networks: A new deep architecture. International conference on computer vision (ICCV), 689–690.
20. RaissiMand Karniadakis GE (2018) Hidden physics models: machine learning of nonlinear partial differential equations. J Comput Phys 357: 125–141.
21. Ray D and Hesthaven JS (2018) An artificial neural network as a troubled cell indicator. J Comput Phys to appear.
22. Ruder S (2017) An overview of gradient descent optimization algorithms. Preprint, arXiv.1609.04747v2.
23. Schwab C and Zech J (2017) Deep learning in high dimension. Technical Report 201757, Seminar for Applied Mathematics, ETH Zürich.
24. Stuart AM (2010) Inverse problems: a Bayesian perspective. Acta Numerica 19: 451–559.
25. Tadmor E (2003) Entropy stability theory for difference approximations of nonlinear conservation laws and related timedependent problems. Acta Numerica, 451–512.
26. Tompson J, Schlachter K, Sprechmann P, et al. (2017) Accelarating Eulerian fluid simulation with convolutional networks. Preprint, arXiv:1607.03597v6.
27. Trefethen LN (2000) Spectral methods in MATLAB, SIAM.
28. Troltzsch F (2010) Optimal control of partial differential equations. AMS.
29. Quateroni A, Manzoni A and Negri F (2015) Reduced basis methods for partial differential equations: an introduction, Springer Verlag.
30. Yarotsky D (2017) Error bounds for approximations with deep ReLU networks. Neural Networks 94: 103–114
© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)