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| Physics-informed Neural Networks for solving convection-diffusion equations on surfaces |
| Received:October 21, 2021 Revised:December 21, 2021 |
| View Full Text View/Add Comment Download reader |
| DOI：10.7511/jslx20211021002 |
| KeyWord:machine learning automatic differentiation Laplace-Beltrami operator physical model surfaces |
| Author | Institution |
| 汤卓超 |
河海大学 力学与材料学院 工程与科学数值模拟软件中心, 南京 |
| 傅卓佳 |
河海大学 力学与材料学院 工程与科学数值模拟软件中心, 南京 |
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| Abstract: |
| This paper introduces Physics-informed Neural Networks and applies them to surface partial differential equations.Different from the traditional Neural Network model,PINNs introduce an automatic differentiation technology in the process of establishing the model and encoding the physical information.By defining the loss function,the optimization goal of the neural network parameters in the model including weightings and bias can be obtained explicitly and solved by existing optimization algorithms.Obviously,PINNs reduce the requirements for the amount of data by adding additional physical information constraints,and show better robustness for some deterministic models.Based on the analytical relationship between surface differential operators and standard differential operators in Euclidean space,this paper introduces two techniques including the extrinsic way and embedding way and combines them with PINNs to solve the convection-diffusion equations defined on high-dimensional complex surfaces.Numerous numerical examples prove the effectiveness,robustness and potential of this method in solving such problems. |