In this paper, we introduce a new deep learning framework for discovering the phase-feld models from existing image data. The new framework embraces the approximation power of physics informed neural networks (PINNs) and the computational efciency of the pseudo-spectral methods, which we named pseudo-spectral PINN or SPINN. Unlike the baseline PINN, the pseudo-spectral PINN has several advantages. First of all, it requires less training data. A minimum of two temporal snapshots with uniform spatial resolution would be adequate. Secondly, it is computationally efcient, as the pseudo-spectral method is used for spatial discretization. Thirdly, it requires less trainable parameters compared with the baseline PINN, which signifcantly simplifes the training process and potentially assures fewer local minima or saddle points. We illustrate the efectiveness of pseudo-spectral PINN through several numerical examples. The newly proposed pseudo-spectral PINN is rather general, and it can be readily applied to discover other PDE-based models from image data.
Jia Zhao
. Discovering Phase Field Models from Image Data with the Pseudo-Spectral Physics Informed Neural Networks[J]. Communications on Applied Mathematics and Computation, 2021
, 3(2)
: 357
-369
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DOI: 10.1007/s42967-020-00105-2
1. Berg, J., Nystrom, K.:A unifed deep artifcial neural network approach to partial diferential equations in complex geometries. Neurocomputing 317, 28-41 (2018)
2. Brunton, S.L., Proctor, J.L., Kutz, J.N.:Discovering governing equations from data by sparse identifcation of nonlinear dynamical systems. Proc. Natl. Acad. Sci. 113(15), 3932-3937 (2016)
3. Cahn, J.W., Hilliard, J.E.:Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258-267 (1958)
4. Chen, L., Zhao, J., Gong, Y.:A novel second-order scheme for the molecular beam epitaxy model with slope selection. Commun. Comput. Phys. 4(25), 1024-1044 (2019)
5. E, Weinan., Yu, B.:The deep Ritz method:a deep learning-based numerical algorithm for solving variational problems. Commun. Math. Stat. 6, 1-12 (2018)
6. Guillen-Gonzalez, F., Tierra, G.:Second order schemes and time-step adaptivity for Allen-Cahn and CahnHilliard models. Comput. Math. Appl. 68(8), 821-846 (2014)
7. Han, D., Wang, X.:A second order in time uniquely solvable unconditionally stable numerical schemes for Cahn-Hilliard-Navier-Stokes equation. J. Comput. Phys. 290(1), 139-156 (2015)
8. Han, J., Jentzen, A., E, Weinan.:Solving high-dimensional partial diferential equations using deep learning. Proc. Natl. Acad. Sci. 115(34), 8505-8510 (2018)
9. Higham, C., Higham, D.:Deep learning:an introduction for applied mathematicians. SIAM Rev. 61(4), 860-891 (2019)
10. Li, B., Tang, S., Yu, H.:Better approximations of high dimensional smooth functions by deep neural networks with rectifed power units. Commun. Comput. Phys. 27, 379-411 (2020)
