VarNet: Variational Neural Networks for the Solution of Partial Differential Equations. In this paper we propose a new model-based unsupervised learning method, called VarNet, for the solution of partial differential equations (PDEs) using deep neural networks (NNs). Particularly, we propose a novel loss function that relies on the variational (integral) form of PDEs as apposed to their differential form which is commonly used in the literature. Our loss function is discretization-free, highly parallelizable, and more effective in capturing the solution of PDEs since it employs lower-order derivatives and trains over measure non-zero regions of space-time. Given this loss function, we also propose an approach to optimally select the space-time samples, used to train the NN, that is based on the feedback provided from the PDE residual. The models obtained using VarNet are smooth and do not require interpolation. They are also easily differentiable and can directly be used for control and optimization of PDEs. Finally, VarNet can straight-forwardly incorporate parametric PDE models making it a natural tool for model order reduction (MOR) of PDEs. We demonstrate the performance of our method through extensive numerical experiments for the advection-diffusion PDE as an important case-study.

References in zbMATH (referenced in 9 articles )

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  1. Gao, Han; Zahr, Matthew J.; Wang, Jian-Xun: Physics-informed graph neural Galerkin networks: a unified framework for solving PDE-governed forward and inverse problems (2022)
  2. Liu, Yujie; Yang, Chao: VPVnet: a velocity-pressure-vorticity neural network method for the Stokes’ equations under reduced regularity (2022)
  3. Rivera, Jon A.; Taylor, Jamie M.; Omella, Ángel J.; Pardo, David: On quadrature rules for solving partial differential equations using neural networks (2022)
  4. Uriarte, Carlos; Pardo, David; Omella, Ángel Javier: A finite element based deep learning solver for parametric PDEs (2022)
  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. Kharazmi, Ehsan; Zhang, Zhongqiang; Karniadakis, George E. M.: \textithp-VPINNs: variational physics-informed neural networks with domain decomposition (2021)
  7. Ranade, Rishikesh; Hill, Chris; Pathak, Jay: Discretizationnet: a machine-learning based solver for Navier-Stokes equations using finite volume discretization (2021)
  8. Rixner, Maximilian; Koutsourelakis, Phaedon-Stelios: A probabilistic generative model for semi-supervised training of coarse-grained surrogates and enforcing physical constraints through virtual observables (2021)
  9. Bao, Gang; Ye, Xiaojing; Zang, Yaohua; Zhou, Haomin: Numerical solution of inverse problems by weak adversarial networks (2020)