KRIZHEVSKY A,SUTSKEVER I,HINTON G E.Imagenet classification with deep convolutional neural networks[J].Advances in Neural Information Processing Systems,2012,25:1097-1105.

LAKE B M,SALAKHUTDINOV R,TENENBAUM J B.Human-level concept learning through probabilistic program induction[J].Science,2015,350:1332-1338.

LECUN Y,BENGIO Y.Convolutional networks for images,speech,and time series[J].The Handbook of Brain Theory and Neural Networks,1995,3361(10):1995.

RAISSI M,PERDIKARIS P,KARNIADAKIS G E.Physics-informed neural networks:a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations[J].Journal of Computational Physics,2019,378:686-707.

TUKEZI A,ABUDUREXITI A.The study of a new numerical method for parabolic partial differential equations[J].Journal of Xinjiang University(Natural Science Edition),2014,31(1):64-69.

JIN X W,CAI S Z,LI H,et al.NSFnets(Navier-Stokes Flow nets):physics-informed neural networks for the incompressible Navier-Stokes equations[J].Joural of Computational Physics,2021,426:109951.

DWIVEDI V,PARASHAR N,SRINIVASAN B.Distributed physics informed neural network for data-efficient solution to partial differential equations[J].ar Xiv preprint ar Xiv,2019,https://doi.org/10.48550/ar Xiv.1907.08967.

KHARAZMI E,ZHANG Z Q,KARNIADAKIS G E.Variational physics-informed neural networks for solving partial differential equations[J].ar Xiv preprint ar Xiv,2019,https://doi.org/10.48550/ar Xiv.1912.00873.

WANG X J,JIA H E.A two-grid method with backtracking for the incompressible Brinkman-Forchheimer equations[J].Journal of Xinjiang University(Natural Science Edition in Chinese and English),2021,38(3):275-284.

BAYDIN A G,PEARLMUTTER B A,RADUL A A,et al.Automatic differentiation in machine learning:a survey[J].The Journal of Machine Learning Research,2018,18:1-43.

WANG S F,TENG Y J,PERDIKARIS P.Understanding and mitigating gradient pathologies in physics-informed neural networks[J].SIAM Journal on Science Computing,2021,43(5):3055-3081.

WU J L,GUI D W,LIU D M,et al.The characteristic variational multiscale method for time dependent conduction-convection problems[J].International Communications in Heat and Mass Transfer,2015,68:58-68.