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We present a methodology combining neural networks with physical principle constraints in the form of partial differential equations (PDEs).
1902
Earlier work this paper cites.
1910
Earlier work this paper cites.
1912
Earlier work this paper cites.
R. Löhner, H. Antil, Revisiting Calderon’s Problem, arXiv:1912.02970
1912
Earlier work this paper cites.
doi:10.1113/jphysiol.1952.sp004764
A. L. Hodgkin, A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of physiology 117 (4) (1952) 500–544 · 1952
Earlier work this paper cites.
D. Noble, A modification of the hodgkin—huxley equations applicable to purkinje fibre action and pacemaker potentials, The Journal of physiology 160 (2) (1962) 317–352
1962
Earlier work this paper cites.
doi:10.1137/0721052
S. G. Nash, Newton-type minimization via the lanczos method, SIAM Journal on Numerical Analysis 21 (4) (1984) 770–788 · 1984
Earlier work this paper cites.
doi:10.1007/BF01589116
D. C. Liu, J. Nocedal, On the limited memory bfgs method for large scale optimization, Mathematical programming 45 (1) (1989) 503–528 · 1989
Earlier work this paper cites.
doi:10.1145/279232.279236
C. Zhu, R. H. Byrd, P. Lu, J. Nocedal, Algorithm 778: L-BFGS-B: Fortran Subroutines for Large-Scale Bound-Constrained Optimization, ACM Trans. Math. Softw. 23 (4) (1997) 550–560 · 1997
Earlier work this paper cites.
doi:10.1016/S0898-1221(99)00248-5
S.-Y. Shen, A numerical study of inverse heat conduction problems, Computers & Mathematics with applications 38 (7-8) (1999) 173–188 · 1999
Earlier work this paper cites.
2001
Earlier work this paper cites.
2001
Earlier work this paper cites.
doi:10.1016/S1570-8659(03)09003-3
R. Glowinski, Finite element methods for incompressible viscous flow, in: Numerical Methods for Fluids (Part 3), Vol. 9 of Handbook of Numerical Analysis, Elsevier, 2003, pp. 3 – 1176 · 2003
Earlier work this paper cites.
2004
Earlier work this paper cites.
doi:10.1023/B:BITN.0000046811.70614.38
A. Kværnø, Singly Diagonally Implicit Runge–Kutta Methods with an Explicit First Stage, BIT Numerical Mathematics 44 (3) (2004) 489–502 · 2004
Earlier work this paper cites.
2005
Cited alongside, same era.
2006
Cited alongside, same era.
2007
Cited alongside, same era.
doi:10.1007/3-540-33437-8
J. Sundnes, G. Lines, X. Cai, B. Nielsen, K. Mardal, A. Tveito, Computing the Electrical Activity in the Heart, Monographs in Computational Science and Engineering, Springer Berlin Heidelberg, 2007 · 2007
Cited alongside, same era.
doi:10.1016/j.jcp.2018.10.045
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 · 2018
Later among the works it cites.
M. Raissi, Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations, Journal of Machine Learning Research 19 (25) (2018) 1–24
2018
Later among the works it cites.
doi:10.1016/j.jcp.2019.108925
Z. Long, Y. Lu, B. Dong, PDE-Net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network, Journal of Computational Physics 399 (2019) 108925 · 2019
Later among the works it cites.
L. Ruthotto, E. Haber, Deep neural networks motivated by partial differential equations, Journal of Mathematical Imaging and Vision (2019) 1–13 doi:10.1007/s10851-019-00903-1
2019
Later among the works it cites.
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2008
Cited alongside, same era.
doi:10.1007/978-1-4020-8839-1_1
S. Ulbrich, Analytical Background and Optimality Theory, Springer Netherlands, Dordrecht, 2009, pp. 1–95 · 2009
Cited alongside, same era.
2010
Cited alongside, same era.
doi:10.1145/1731022.1731030
A. Logg, G. N. Wells, DOLFIN: Automated finite element computing, ACM Transactions on Mathematical Software (TOMS) 37 (2) (2010) 20:1–20:28 · 2010
Cited alongside, same era.
2011
Cited alongside, same era.
U. Naumann, The Art of Differentiating Computer Programs: An Introduction to Algorithmic Differentiation, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2012
2012
Cited alongside, same era.
K. E. Skare, Gryphon-a module for time integration of partial differential equations in fenics, Master’s thesis, Institutt for matematiske fag (2012)
2012
Cited alongside, same era.
doi:10.1145/2566630
M. S. Alnæs, A. Logg, K. B. Ølgaard, M. E. Rognes, G. N. Wells, Unified Form Language: A Domain-specific Language for Weak Formulations of Partial Differential Equations, ACM Trans. Math. Softw. 40 (2) (2014) 9:1–9:37 · 2014
Cited alongside, same era.
A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, S. Chintala, Pytorch: An imperative style, high-performance deep learning library, in: Advances in Neural Information Processing Systems 32, Curran Associates, Inc., 2019, pp. 8024–8035
2019
Later among the works it cites.
K. Duraisamy, G. Iaccarino, H. Xiao, Turbulence modeling in the age of data, Annual Review of Fluid Mechanics 51 (1) (2019) 357–377
2019
Later among the works it cites.
doi:10.21105/joss.01292
S. K. Mitusch, S. W. Funke, J. S. Dokken, dolfin-adjoint 2018.1: automated adjoints for FEniCS and Firedrake, Journal of Open Source Software 4 (38) (2019) 1292 · 2019
Later among the works it cites.
doi:10.1016/j.jcp.2019.109136
A. D. Jagtap, K. Kawaguchi, G. E. Karniadakis, Adaptive activation functions accelerate convergence in deep and physics-informed neural networks, Journal of Computational Physics 404 (2020) 109136 · 2019
Later among the works it cites.
doi:10.1137/17M1144532
P. E. Farrell, J. E. Hake, S. W. Funke, M. E. Rognes, Automated adjoints of coupled PDE-ODE systems, SIAM Journal on Scientific Computing 41 (3) (2019) C219–C244 · 2019
Later among the works it cites.
doi:10.1016/j.jcp.2020.109491
D. Z. Huang, K. Xu, C. Farhat, E. Darve, Learning Constitutive Relations from Indirect Observations Using Deep Neural Networks, Journal of Computational Physics 416 (2020) 109491 · 2020
Later among the works it cites.
doi:10.1038/s41592-019-0686-2
P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, SciPy 1.0 Contributors, SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python, Nature Methods 17 (2020) 261–272 · 2020
Later among the works it cites.
L. M. Valnes, S. K. Mitusch, G. Ringstad, P. K. Eide, S. W. Funke, K.-A. Mardal, Apparent diffusion coefficient estimates based on 24 hours tracer movement support glymphatic transport in human cerebral cortex, Scientific Reports 10 (1) (2020) 1–12
2020
Later among the works it cites.
doi:10.1016/j.cma.2020.113028
A. D. Jagtap, E. Kharazmi, G. E. Karniadakis, Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems, Computer Methods in Applied Mechanics and Engineering 365 (2020) 113028 · 2020
Later among the works it cites.
A. D. Jagtap, G. E. Karniadakis, Extended Physics-Informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition Based Deep Learning Framework for Nonlinear Partial Differential Equations, Communications in Computational Physics (2020) 2002–2041 doi:10.4208/cicp.OA-2020-0164
2020
Later among the works it cites.