In this paper,the physics informed neural network(PINN)deep learning method is applied to solve two-dimensional nonlocal equations,including the partial reverse space y-nonlocal Mel'nikov equation,the partial reve...In this paper,the physics informed neural network(PINN)deep learning method is applied to solve two-dimensional nonlocal equations,including the partial reverse space y-nonlocal Mel'nikov equation,the partial reverse space-time nonlocal Mel'nikov equation and the nonlocal twodimensional nonlinear Schr?dinger(NLS)equation.By the PINN method,we successfully derive a data-driven two soliton solution,lump solution and rogue wave solution.Numerical simulation results indicate that the error range between the data-driven solution and the exact solution is relatively small,which verifies the effectiveness of the PINN deep learning method for solving high dimensional nonlocal equations.Moreover,the parameter discovery of the partial reverse space-time nonlocal Mel'nikov equation is analysed in terms of its soliton solution for the first time.展开更多
本文采用物理信息神经网络(PINN)来求解不可压缩湍流Navier-Stokes方程。本研究引入了动态权重调整策略,使得各项误差在训练过程中得到适当的平衡,从而避免了某些误差项主导整个训练过程的问题。此外,为了加速训练收敛并提高精度,本研...本文采用物理信息神经网络(PINN)来求解不可压缩湍流Navier-Stokes方程。本研究引入了动态权重调整策略,使得各项误差在训练过程中得到适当的平衡,从而避免了某些误差项主导整个训练过程的问题。此外,为了加速训练收敛并提高精度,本研究还对网络结构进行了优化,结合物理约束优化过程,改变了优化方法,提高了模型的训练效率。In this paper, physical information neural networks (PINN) are used to solve the Navier-Stokes equations of incompressible turbulence. In this study, the dynamic weighting adjustment strategy is presented to make the errors properly balanced in the training process, so as to avoid the problem that some error terms dominate the whole training process. In addition, in order to accelerate the training convergence and improve the accuracy, this study also optimized the network structure, combining with the physical constraint optimization process and changing the optimization method to improve the training efficiency of the model.展开更多
This is the second part of our series works on failure-informed adaptive sampling for physic-informed neural networks(PINNs).In our previous work(SIAM J.Sci.Comput.45:A1971–A1994),we have presented an adaptive sampli...This is the second part of our series works on failure-informed adaptive sampling for physic-informed neural networks(PINNs).In our previous work(SIAM J.Sci.Comput.45:A1971–A1994),we have presented an adaptive sampling framework by using the failure probability as the posterior error indicator,where the truncated Gaussian model has been adopted for estimating the indicator.Here,we present two extensions of that work.The first extension consists in combining with a re-sampling technique,so that the new algorithm can maintain a constant training size.This is achieved through a cosine-annealing,which gradually transforms the sampling of collocation points from uniform to adaptive via the training progress.The second extension is to present the subset simulation(SS)algorithm as the posterior model(instead of the truncated Gaussian model)for estimating the error indicator,which can more effectively estimate the failure probability and generate new effective training points in the failure region.We investigate the performance of the new approach using several challenging problems,and numerical experiments demonstrate a significant improvement over the original algorithm.展开更多
In this paper, we use Physics-Informed Neural Networks (PINNs) to solve shape optimization problems. These problems are based on incompressible Navier-Stokes equations and phase-field equations. The phase-field functi...In this paper, we use Physics-Informed Neural Networks (PINNs) to solve shape optimization problems. These problems are based on incompressible Navier-Stokes equations and phase-field equations. The phase-field function is used to describe the state of the fluids, and the optimal shape optimization is obtained by using the shape sensitivity analysis based on the phase-field function. The sharp interface is also presented by a continuous function between zero and one with a large gradient. To avoid the numerical solutions falling into the trivial solution, the hard boundary condition is implemented for our PINNs’ training. Finally, numerical results are given to prove the feasibility and effectiveness of the proposed numerical method.展开更多
文摘In this paper,the physics informed neural network(PINN)deep learning method is applied to solve two-dimensional nonlocal equations,including the partial reverse space y-nonlocal Mel'nikov equation,the partial reverse space-time nonlocal Mel'nikov equation and the nonlocal twodimensional nonlinear Schr?dinger(NLS)equation.By the PINN method,we successfully derive a data-driven two soliton solution,lump solution and rogue wave solution.Numerical simulation results indicate that the error range between the data-driven solution and the exact solution is relatively small,which verifies the effectiveness of the PINN deep learning method for solving high dimensional nonlocal equations.Moreover,the parameter discovery of the partial reverse space-time nonlocal Mel'nikov equation is analysed in terms of its soliton solution for the first time.
文摘本文采用物理信息神经网络(PINN)来求解不可压缩湍流Navier-Stokes方程。本研究引入了动态权重调整策略,使得各项误差在训练过程中得到适当的平衡,从而避免了某些误差项主导整个训练过程的问题。此外,为了加速训练收敛并提高精度,本研究还对网络结构进行了优化,结合物理约束优化过程,改变了优化方法,提高了模型的训练效率。In this paper, physical information neural networks (PINN) are used to solve the Navier-Stokes equations of incompressible turbulence. In this study, the dynamic weighting adjustment strategy is presented to make the errors properly balanced in the training process, so as to avoid the problem that some error terms dominate the whole training process. In addition, in order to accelerate the training convergence and improve the accuracy, this study also optimized the network structure, combining with the physical constraint optimization process and changing the optimization method to improve the training efficiency of the model.
基金supported by the NSF of China(No.12171085)This work was supported by the National Key R&D Program of China(2020YFA0712000)+2 种基金the NSF of China(No.12288201)the Strategic Priority Research Program of Chinese Academy of Sciences(No.XDA25010404)and the Youth Innovation Promotion Association(CAS).
文摘This is the second part of our series works on failure-informed adaptive sampling for physic-informed neural networks(PINNs).In our previous work(SIAM J.Sci.Comput.45:A1971–A1994),we have presented an adaptive sampling framework by using the failure probability as the posterior error indicator,where the truncated Gaussian model has been adopted for estimating the indicator.Here,we present two extensions of that work.The first extension consists in combining with a re-sampling technique,so that the new algorithm can maintain a constant training size.This is achieved through a cosine-annealing,which gradually transforms the sampling of collocation points from uniform to adaptive via the training progress.The second extension is to present the subset simulation(SS)algorithm as the posterior model(instead of the truncated Gaussian model)for estimating the error indicator,which can more effectively estimate the failure probability and generate new effective training points in the failure region.We investigate the performance of the new approach using several challenging problems,and numerical experiments demonstrate a significant improvement over the original algorithm.
文摘In this paper, we use Physics-Informed Neural Networks (PINNs) to solve shape optimization problems. These problems are based on incompressible Navier-Stokes equations and phase-field equations. The phase-field function is used to describe the state of the fluids, and the optimal shape optimization is obtained by using the shape sensitivity analysis based on the phase-field function. The sharp interface is also presented by a continuous function between zero and one with a large gradient. To avoid the numerical solutions falling into the trivial solution, the hard boundary condition is implemented for our PINNs’ training. Finally, numerical results are given to prove the feasibility and effectiveness of the proposed numerical method.
