fPINNs: Fractional Physics-Informed Neural Networks. Physics-informed neural networks (PINNs) are effective in solving integer-order partial differential equations (PDEs) based on scattered and noisy data. PINNs employ standard feedforward neural networks (NNs) with the PDEs explicitly encoded into the NN using automatic differentiation, while the sum of the mean-squared PDE-residuals and the mean-squared error in initial/boundary conditions is minimized with respect to the NN parameters. We extend PINNs to fractional PINNs (fPINNs) to solve space-time fractional advection-diffusion equations (fractional ADEs), and we demonstrate their accuracy and effectiveness in solving multi-dimensional forward and inverse problems with forcing terms whose values are only known at randomly scattered spatio-temporal coordinates (black-box forcing terms). A novel element of the fPINNs is the hybrid approach that we introduce for constructing the residual in the loss function using both automatic differentiation for the integer-order operators and numerical discretization for the fractional operators. We consider 1D time-dependent fractional ADEs and compare white-box (WB) and black-box (BB) forcing. We observe that for the BB forcing fPINNs outperform FDM. Subsequently, we consider multi-dimensional time-, space-, and space-time-fractional ADEs using the directional fractional Laplacian and we observe relative errors of 10−4. Finally, we solve several inverse problems in 1D, 2D, and 3D to identify the fractional orders, diffusion coefficients, and transport velocities and obtain accurate results even in the presence of significant noise

References in zbMATH (referenced in 27 articles , 1 standard article )

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  1. Li, Yixin; Hu, Xianliang: Artificial neural network approximations of Cauchy inverse problem for linear PDEs (2022)
  2. Mo, Yifan; Ling, Liming; Zeng, Delu: Data-driven vector soliton solutions of coupled nonlinear Schrödinger equation using a deep learning algorithm (2022)
  3. Rudin, Cynthia; Chen, Chaofan; Chen, Zhi; Huang, Haiyang; Semenova, Lesia; Zhong, Chudi: Interpretable machine learning: fundamental principles and 10 grand challenges (2022)
  4. Wang, Hengjie; Planas, Robert; Chandramowlishwaran, Aparna; Bostanabad, Ramin: Mosaic flows: a transferable deep learning framework for solving PDEs on unseen domains (2022)
  5. Camilli, Fabio; Duisembay, Serikbolsyn; Tang, Qing: Approximation of an optimal control problem for the time-fractional Fokker-Planck equation (2021)
  6. Chen, Xiaoli; Duan, Jinqiao; Karniadakis, George Em: Learning and meta-learning of stochastic advection-diffusion-reaction systems from sparse measurements (2021)
  7. Chen, Xiaoli; Yang, Liu; Duan, Jinqiao; Karniadakis, George Em: Solving inverse stochastic problems from discrete particle observations using the Fokker-Planck equation and physics-informed neural networks (2021)
  8. D’Elia, Marta; Gunzburger, Max; Vollmann, Christian: A cookbook for approximating Euclidean balls and for quadrature rules in finite element methods for nonlocal problems (2021)
  9. D’Elia, M.; De Los Reyes, J. C.; Miniguano-Trujillo, A.: Bilevel parameter learning for nonlocal image denoising models (2021)
  10. Kharazmi, Ehsan; Zhang, Zhongqiang; Karniadakis, George E. M.: \textithp-VPINNs: variational physics-informed neural networks with domain decomposition (2021)
  11. Lu, Lu; Meng, Xuhui; Mao, Zhiping; Karniadakis, George Em: DeepXDE: a deep learning library for solving differential equations (2021)
  12. Lu, Lu; Pestourie, Raphaël; Yao, Wenjie; Wang, Zhicheng; Verdugo, Francesc; Johnson, Steven G.: Physics-informed neural networks with hard constraints for inverse design (2021)
  13. Nguyen-Thanh, Vien Minh; Anitescu, Cosmin; Alajlan, Naif; Rabczuk, Timon; Zhuang, Xiaoying: Parametric deep energy approach for elasticity accounting for strain gradient effects (2021)
  14. Wang, Li; Yan, Zhenya: Data-driven peakon and periodic peakon solutions and parameter discovery of some nonlinear dispersive equations via deep learning (2021)
  15. Yin, Minglang; Zheng, Xiaoning; Humphrey, Jay D.; Karniadakis, George Em: Non-invasive inference of thrombus material properties with physics-informed neural networks (2021)
  16. You, Huaiqian; Yu, Yue; Trask, Nathaniel; Gulian, Mamikon; D’Elia, Marta: Data-driven learning of nonlocal physics from high-fidelity synthetic data (2021)
  17. Zhao, Dazhi; Yu, Guozhu; Li, Weibin: Diffusion on fractal objects modeling and its physics-informed neural network solution (2021)
  18. Zhou, Zijian; Yan, Zhenya: Solving forward and inverse problems of the logarithmic nonlinear Schrödinger equation with (\mathcalPT)-symmetric harmonic potential via deep learning (2021)
  19. Zhuang, Xiaoying; Guo, Hongwei; Alajlan, Naif; Zhu, Hehua; Rabczuk, Timon: Deep autoencoder based energy method for the bending, vibration, and buckling analysis of Kirchhoff plates with transfer learning (2021)
  20. Darbon, Jérôme; Langlois, Gabriel P.; Meng, Tingwei: Overcoming the curse of dimensionality for some Hamilton-Jacobi partial differential equations via neural network architectures (2020)

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