Fetching the paper…
Reading the bibliography…
We formulate a general framework for hp-variational physics-informed neural networks (hp-VPINNs) based on the nonlinear approximation of shallow and deep neural networks and hp-refinement via domain decomposition and projection onto space of high-order polynomials.
1909
Earlier work this paper cites.
S. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Soviet Math. Dokl. 4 (1963) 240–243
1963
Earlier work this paper cites.
B. A. Finlayson, L. E. Scriven, The method of weighted residuals—A review, Applied Mechanics Review 19 (9) (1966) 735–748
1966
Earlier work this paper cites.
I. Daubechies, Ten lectures on wavelets, Vol. 61, SIAM, 1992
1992
Earlier work this paper cites.
H. N. Mhaskar, C. A. Micchelli, Approximation by superposition of sigmoidal and radial basis functions, Advances in Applied Mathematics 13 (3) (1992) 350–373
1992
Earlier work this paper cites.
G. Davis, Adaptive nonlinear approximations, Ph.D. thesis, New York University, Graduate School of Arts and Science (1994)
1994
Earlier work this paper cites.
W. J. Morokoff, R. E. Caflisch, Quasi-Monte Carlo integration, Journal of Computational Physics 122 (2) (1995) 218–230
1995
Earlier work this paper cites.
E. Novak, K. Ritter, High dimensional integration of smooth functions over cubes, Numerische Mathematik 75 (1) (1996) 79–97
1996
Earlier work this paper cites.
R. A. DeVore, Nonlinear approximation, Acta Numerica 7 (1998) 51–150
1998
Earlier work this paper cites.
E. J. Candès, et al., Compressive sampling, in: Proceedings of the International Congress of Mathematicians, Vol. 3, Madrid, Spain, 2006, pp. 1433–1452
2006
Earlier work this paper cites.
E. J. Candès, M. B. Wakin, An introduction to compressive sampling [a sensing/sampling paradigm that goes against the common knowledge in data acquisition], IEEE Signal Processing Magazine 25 (2) (2008) 21–30
2008
Earlier work this paper cites.
R. A. DeVore, Nonlinear approximation and its applications, in: Multiscale, Nonlinear and Adaptive Approximation, Springer, 2009, pp. 169–201
2009
Earlier work this paper cites.
R. DeVore, A. Ron, Approximation using scattered shifts of a multivariate function, Transactions of the American Mathematical Society 362 (12) (2010) 6205–6229
2010
Earlier work this paper cites.
T. Hangelbroek, A. Ron, Nonlinear approximation using Gaussian kernels, Journal of Functional Analysis 259 (1) (2010) 203–219
2010
Earlier work this paper cites.
H. Ohlsson, A. Y. Yang, R. Dong, S. S. Sastry, Nonlinear basis pursuit, in: 2013 Asilomar Conference on Signals, Systems and Computers, IEEE, 2013, pp. 115–119
2013
Earlier work this paper cites.
G. E. Karniadakis, S. J. Sherwin, Spectral/ h p hp Element Methods for Computational Fluid Dynamics, Oxford University Press, New York, 2013
2013
Earlier work this paper cites.
2014
Cited alongside, same era.
A. Mojtabi, M. O. Deville, One-dimensional linear advection–diffusion equation: Analytical and finite element solutions, Computers & Fluids 107 (2015) 189–195
2015
Cited alongside, same era.
2016
Cited alongside, same era.
M. Raissi, P. Perdikaris, G. E. Karniadakis, Machine learning of linear differential equations using Gaussian processes, Journal of Computational Physics 348 (2017) 683–693
2017
Cited alongside, same era.
doi:https://doi.org/10.1016/j.neucom.2018.06.056
J. Berg, K. Nyström, A unified deep artificial neural network approach to partial differential equations in complex geometries , Neurocomputing 317 (2018) 28 – 41 · 2018
2019
Later among the works it cites.
2019
Later among the works it cites.
M. Raissi, H. Babaee, P. Givi, Deep learning of turbulent scalar mixing, Physical Review Fluids 4 (12) (2019) 124501
2019
Later among the works it cites.
G. Pang, L. Lu, G. E. Karniadakis, fPINNs: Fractional physics-informed neural networks, SIAM Journal on Scientific Computing 41 (4) (2019) A2603–A2626
2019
Later among the works it cites.
A. D. Jagtap, K. Kawaguchi, G. E. Karniadakis, Adaptive activation functions accelerate convergence in deep and physics-informed neural networks, Journal of Computational Physics (2019) 109–136
alphaXiv searches the wider corpus for related work and actual follow-ups.
alphaXiv is searching for related work…
Cited alongside, same era.
W. E, B. Yu, The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems, Communications in Mathematics and Statistics 6 (1) (2018) 1–12
2018
Cited alongside, same era.
J. Sirignano, K. Spiliopoulos, DGM: A deep learning algorithm for solving partial differential equations, Journal of Computational Physics 375 (2018) 1339–1364
2018
Cited alongside, same era.
2018
Cited alongside, same era.
2018
Cited alongside, same era.
2018
Cited alongside, same era.
M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics 378 (2019) 686–707
2019
Cited alongside, same era.
Y. Khoo, J. Lu, L. Ying, Solving for high-dimensional committor functions using artificial neural networks, Research in the Mathematical Sciences 6 (1) (2019) 1
2019
Cited alongside, same era.
2019
Later among the works it cites.
2019
Later among the works it cites.
2019
Later among the works it cites.
2019
Later among the works it cites.
2019
Later among the works it cites.
2019
Later among the works it cites.
Z. Mao, A. D. Jagtap, G. E. Karniadakis, Physics-informed neural networks for high-speed flows, Computer Methods in Applied Mechanics and Engineering 360 (2020) 112789
2020
Closest in time.
E. Samaniego, C. Anitescu, S. Goswami, V. M. Nguyen-Thanh, H. Guo, K. Hamdia, X. Zhuang, T. Rabczuk, An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, Implementation and Applications, Computer Methods in Applied Mechanics and Engineering 362 (2020) 112790
2020
Closest in time.
2020
Closest in time.
2020
Closest in time.
A. D. Jagtap, E. Kharazmi, G. E. Karniadakis, Conservative physics-informed neural networks on discrete domains for conservation laws, Computer Methods in Applied Mechanics and Engineering - under revision (2020)
2020
Closest in time.