D3M: A deep domain decomposition method for partial differential equations. A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs). The solution of PDEs can be formulated as the solution of a constrained optimization problem, and we design a multi-fidelity neural network framework to solve this optimization problem. Our contribution is to develop a systematical computational procedure for the underlying problem in parallel with domain decomposition. Our analysis shows that the D3M approximation solution converges to the exact solution of underlying PDEs. Our proposed framework establishes a foundation to use variational deep learning in large-scale engineering problems and designs. We present a general mathematical framework of D3M, validate its accuracy and demonstrate its efficiency with numerical experiments.

References in zbMATH (referenced in 11 articles )

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  1. Dong, Suchuan; Yang, Jielin: On computing the hyperparameter of extreme learning machines: algorithm and application to computational PDEs, and comparison with classical and high-order finite elements (2022)
  2. Lyu, Liyao; Zhang, Zhen; Chen, Minxin; Chen, Jingrun: MIM: a deep mixed residual method for solving high-order partial differential equations (2022)
  3. Tang, Kejun; Wan, Xiaoliang; Liao, Qifeng: Adaptive deep density approximation for Fokker-Planck equations (2022)
  4. Dong, Suchuan; Li, Zongwei: Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations (2021)
  5. Gao, Han; Sun, Luning; Wang, Jian-Xun: PhyGeoNet: physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain (2021)
  6. Gu, Yiqi; Yang, Haizhao; Zhou, Chao: SelectNet: self-paced learning for high-dimensional partial differential equations (2021)
  7. Kharazmi, Ehsan; Zhang, Zhongqiang; Karniadakis, George E. M.: \textithp-VPINNs: variational physics-informed neural networks with domain decomposition (2021)
  8. Ranade, Rishikesh; Hill, Chris; Pathak, Jay: Discretizationnet: a machine-learning based solver for Navier-Stokes equations using finite volume discretization (2021)
  9. Sheng, Hailong; Yang, Chao: PFNN: a penalty-free neural network method for solving a class of second-order boundary-value problems on complex geometries (2021)
  10. Jagtap, Ameya D.; Karniadakis, George Em: Extended physics-informed neural networks (XPINNs): a generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations (2020)
  11. Jagtap, Ameya D.; Kharazmi, Ehsan; Karniadakis, George Em: Conservative physics-informed neural networks on discrete domains for conservation laws: applications to forward and inverse problems (2020)