Adaptive activation functions accelerate convergence in deep and physics-informed neural networks

Jagtap, Ameya D.; Kawaguchi, Kenji; Karniadakis, George Em

doi:10.1016/j.jcp.2019.109136

Adaptive activation functions accelerate convergence in deep and physics-informed neural networks

Journal Article · Sun Nov 24 23:00:00 EST 2019 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2019.109136· OSTI ID:2282002

^[1]; Kawaguchi, Kenji ^[2]; Karniadakis, George Em ^[3]

Brown Univ., Providence, RI (United States); Brown University
Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
Brown Univ., Providence, RI (United States); Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)

Here 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. Furthermore, the proposed adaptive activation functions are shown to accelerate the minimization process of the loss values in standard deep learning benchmarks using CIFAR-10, CIFAR-100, SVHN, MNIST, KMNIST, Fashion-MNIST, and Semeion datasets with and without data augmentation.

View Accepted Manuscript (DOE)

Research Organization:: Brown Univ., Providence, RI (United States)

Sponsoring Organization:: USDOE; Defense Advanced Research Projects Agency (DARPA)

Grant/Contract Number:: SC0019453

OSTI ID:: 2282002

Alternate ID(s):: OSTI ID: 1617451
OSTI ID: 1775904

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 404; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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