Adaptive Activation Functions Accelerate Convergence in Deep and Physics-informed Neural Networks

Jagtap, Ameya; Kawaguchi, Kenji; Karniadakis, George E.

doi:10.1016/j.jcp.2019.109136

Adaptive Activation Functions Accelerate Convergence in Deep and Physics-informed Neural Networks

Journal Article · Sat Feb 29 23:00:00 EST 2020 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2019.109136· OSTI ID:1617451

Jagtap, Ameya ^[1]; Kawaguchi, Kenji ^[2]; Karniadakis, George E. ^[3]

Brown University
Massachusetts Institute of Technology
BROWN UNIVERSITY

We employ adaptive activation functions for regression in deep and physics-informed neural networks (PINNs) to approximate smooth and discontinuous functions as well as solutions of linear and nonlinear partial differential equations. In particular, we solve the nonlinear Klein-Gordon equation, which has smooth solutions, the nonlinear Burgers equation, which can admit high gradient solutions, and the Helmholtz equation. We introduce a scalable hyper-parameter in the activation function, which can be optimized to achieve best performance of the network as it changes dynamically the topology of the loss function involved in the optimization process. The adaptive activation function has better learning capabilities than the traditional one (fixed activation) as it improves greatly the convergence rate, especially at early training, as well as the solution accuracy. To better understand the learning process, we plot the neural network solution in the frequency domain to examine how the network captures successively different frequency bands present in the solution. We consider both forward problems, where the approximate solutions are obtained, as well as inverse problems, where parameters involved in the governing equation are identified. Our simulation results show that the proposed method is a very simple and effective approach to increase the efficiency, robustness and accuracy of the neural network approximation of nonlinear functions as well as solutions of partial differential equations, especially for forward problems. We theoretically prove that in the proposed method, gradient descent algorithms are not attracted to suboptimal critical points or local minima.

Research Organization:: Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)

Sponsoring Organization:: USDOE

DOE Contract Number:: AC05-76RL01830

OSTI ID:: 1617451

Report Number(s):: PNNL-SA-152708

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 404

Country of Publication:: United States

Language:: English

References (22)

Automatic differentiation in machine learning: a survey Baydin, Atilim Gunes; Pearlmutter, Barak A.; Radul, Alexey Andreyevich arXiv https://doi.org/10.48550/arxiv.1502.05767	text	January 2015
Machine learning of linear differential equations using Gaussian processes Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em Journal of Computational Physics, Vol. 348 https://doi.org/10.1016/j.jcp.2017.07.050	journal	November 2017
Deep learning of vortex-induced vibrations Raissi, Maziar; Wang, Zhicheng; Triantafyllou, Michael S. Journal of Fluid Mechanics, Vol. 861 https://doi.org/10.1017/jfm.2018.872	journal	December 2018
A New Multi-output Neural Model with Tunable Activation Function and its Applications Shen, Yanjun; Wang, Bingwen; Chen, Fangxin Neural Processing Letters, Vol. 20, Issue 2 https://doi.org/10.1007/s11063-004-0637-4	journal	November 2004
A paradigm for data-driven predictive modeling using field inversion and machine learning Parish, Eric J.; Duraisamy, Karthik Journal of Computational Physics, Vol. 305 https://doi.org/10.1016/j.jcp.2015.11.012	journal	January 2016
The sine-Gordon equation as a model classical field theory Caudrey, P. J.; Eilbeck, J. C.; Gibbon, J. D. Il Nuovo Cimento B Series 11, Vol. 25, Issue 2 https://doi.org/10.1007/BF02724733	journal	February 1975
The extreme learning machine learning algorithm with tunable activation function Li, Bin; Li, Yibin; Rong, Xuewen Neural Computing and Applications, Vol. 22, Issue 3-4 https://doi.org/10.1007/s00521-012-0858-9	journal	February 2012
Data-driven discovery of PDEs in complex datasets Berg, Jens; Nyström, Kaj Journal of Computational Physics, Vol. 384 https://doi.org/10.1016/j.jcp.2019.01.036	journal	May 2019
Artificial neural networks for solving ordinary and partial differential equations Lagaris, I. E.; Likas, A.; Fotiadis, D. I. IEEE Transactions on Neural Networks, Vol. 9, Issue 5 https://doi.org/10.1109/72.712178	journal	January 1998
Some Recent Researches on the Motion of Fluids Bateman, Harry Monthly Weather Review, Vol. 43, Issue 4 https://doi.org/10.1175/1520-0493(1915)43<163:SRROTM>2.0.CO;2	journal	April 1915
Data-driven discovery of partial differential equations Rudy, Samuel H.; Brunton, Steven L.; Proctor, Joshua L. Science Advances, Vol. 3, Issue 4 https://doi.org/10.1126/sciadv.1602614	journal	April 2017
New travelling wave solutions to the Boussinesq and the Klein–Gordon equations Wazwaz, Abdul-Majid Communications in Nonlinear Science and Numerical Simulation, Vol. 13, Issue 5 https://doi.org/10.1016/j.cnsns.2006.08.005	journal	July 2008
Adaptive activation functions in convolutional neural networks Qian, Sheng; Liu, Hua; Liu, Cheng Neurocomputing, Vol. 272 https://doi.org/10.1016/j.neucom.2017.06.070	journal	January 2018
Spectral and finite difference solutions of the Burgers equation Basdevant, C.; Deville, M.; Haldenwang, P. Computers & Fluids, Vol. 14, Issue 1 https://doi.org/10.1016/0045-7930(86)90036-8	journal	January 1986
Inferring solutions of differential equations using noisy multi-fidelity data Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em Journal of Computational Physics, Vol. 335 https://doi.org/10.1016/j.jcp.2017.01.060	journal	April 2017
Gradient-based learning applied to document recognition Lecun, Y.; Bottou, L.; Bengio, Y. Proceedings of the IEEE, Vol. 86, Issue 11 https://doi.org/10.1109/5.726791	journal	January 1998
A Mathematical Model Illustrating the Theory of Turbulence Burgers, J. M. Advances in Applied Mechanics https://doi.org/10.1016/S0065-2156(08)70100-5	book	January 1948
Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations Raissi, M.; Perdikaris, P.; Karniadakis, G. E. Journal of Computational Physics, Vol. 378 https://doi.org/10.1016/j.jcp.2018.10.045	journal	February 2019
Bayesian Numerical Homogenization Owhadi, Houman Multiscale Modeling & Simulation, Vol. 13, Issue 3 https://doi.org/10.1137/140974596	journal	January 2015
Hidden physics models: Machine learning of nonlinear partial differential equations Raissi, Maziar; Karniadakis, George Em Journal of Computational Physics, Vol. 357 https://doi.org/10.1016/j.jcp.2017.11.039	journal	March 2018
Numerical Gaussian Processes for Time-Dependent and Nonlinear Partial Differential Equations Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em SIAM Journal on Scientific Computing, Vol. 40, Issue 1 https://doi.org/10.1137/17M1120762	journal	January 2018
Learning long-term dependencies with gradient descent is difficult Bengio, Y.; Simard, P.; Frasconi, P. IEEE Transactions on Neural Networks, Vol. 5, Issue 2 https://doi.org/10.1109/72.279181	journal	March 1994

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