Neural network methods have been widely used in many fields of scientific research with the rapid increase of computing power.The physics-informed neural networks(PINNs)have received much attention as a major breakthr...Neural network methods have been widely used in many fields of scientific research with the rapid increase of computing power.The physics-informed neural networks(PINNs)have received much attention as a major breakthrough in solving partial differential equations using neural networks.In this paper,a resampling technique based on the expansion-shrinkage point(ESP)selection strategy is developed to dynamically modify the distribution of training points in accordance with the performance of the neural networks.In this new approach both training sites with slight changes in residual values and training points with large residuals are taken into account.In order to make the distribution of training points more uniform,the concept of continuity is further introduced and incorporated.This method successfully addresses the issue that the neural network becomes ill or even crashes due to the extensive alteration of training point distribution.The effectiveness of the improved physics-informed neural networks with expansion-shrinkage resampling is demonstrated through a series of numerical experiments.展开更多
In this work,a new method to solve the Reynolds equation including mass-conserving cavitation by using the physics informed neural networks(PINNs)is proposed.The complementarity relationship between the pressure and t...In this work,a new method to solve the Reynolds equation including mass-conserving cavitation by using the physics informed neural networks(PINNs)is proposed.The complementarity relationship between the pressure and the void fraction is used.There are several difficulties in problem solving,and the solutions are provided.Firstly,the difficulty for considering the pressure inequality constraint by PINNs is solved by transferring it into one equality constraint without introducing error.While the void fraction inequality constraint is considered by using the hard constraint with the max-min function.Secondly,to avoid the fluctuation of the boundary value problems,the hard constraint method is also utilized to apply the boundary pressure values and the corresponding functions are provided.Lastly,for avoiding the trivial solution the limitation for the mean value of the void fraction is applied.The results are validated against existing data,and both the incompressible and compressible lubricant are considered.Good agreement can be found for both the domain and domain boundaries.展开更多
In this paper,we introduce a new deep learning framework for discovering the phase-field models from existing image data.The new framework embraces the approximation power of physics informed neural networks(PINNs)and...In this paper,we introduce a new deep learning framework for discovering the phase-field models from existing image data.The new framework embraces the approximation power of physics informed neural networks(PINNs)and the computational efficiency of the pseudo-spectral methods,which we named pseudo-spectral PINN or SPINN.Unlike the baseline PINN,the pseudo-spectral PINN has several advantages.First of all,it requires less training data.A minimum of two temporal snapshots with uniform spatial resolution would be adequate.Secondly,it is computationally efficient,as the pseudo-spectral method is used for spatial discretization.Thirdly,it requires less trainable parameters compared with the baseline PINN,which significantly simplifies the training process and potentially assures fewer local minima or saddle points.We illustrate the effectiveness of pseudo-spectral PINN through several numerical examples.The newly proposed pseudo-spectral PINN is rather general,and it can be readily applied to discover other FDE-based models from image data.展开更多
传统的数值求解方法面临维数灾难和效率与精度平衡问题,而基于数据驱动的神经网络求解方法又存在训练量冗余和不可解释性问题。针对此问题,物理信息神经网络(Physical Information Neural Networks,PINNs)关注了训练数据中隐含的物理先...传统的数值求解方法面临维数灾难和效率与精度平衡问题,而基于数据驱动的神经网络求解方法又存在训练量冗余和不可解释性问题。针对此问题,物理信息神经网络(Physical Information Neural Networks,PINNs)关注了训练数据中隐含的物理先验知识,融合了神经网络拟合复杂变量的能力,赋予了传统神经网络所缺乏的物理可解释性。应用该算法模型,提出了一种基于PINN的Burgers方程求解模型,该算法模型在训练中施加物理信息约束,因此能用少量的训练样本学习预测到分布在时空域上的偏微分方程模型。实验结果表明,在1+1维Burgers方程算例下,所提方法相比于经典的机器学习算法能有效捕抓到方程的变化并进行精确模拟,相比于有限差分法,可以大幅度缩短模拟时间。通过对不同的网络参数进行比较实验,所提方法在10%的噪声破坏下能产生合理的识别准确度,网络逼近方程的待定系数误差在0.001以内。展开更多
With the advent of physics informed neural networks(PINNs),deep learning has gained interest for solving nonlinear partial differential equations(PDEs)in recent years.In this paper,physics informed memory networks(PIM...With the advent of physics informed neural networks(PINNs),deep learning has gained interest for solving nonlinear partial differential equations(PDEs)in recent years.In this paper,physics informed memory networks(PIMNs)are proposed as a new approach to solving PDEs by using physical laws and dynamic behavior of PDEs.Unlike the fully connected structure of the PINNs,the PIMNs construct the long-term dependence of the dynamics behavior with the help of the long short-term memory network.Meanwhile,the PDEs residuals are approximated using difference schemes in the form of convolution filter,which avoids information loss at the neighborhood of the sampling points.Finally,the performance of the PIMNs is assessed by solving the Kd V equation and the nonlinear Schr?dinger equation,and the effects of difference schemes,boundary conditions,network structure and mesh size on the solutions are discussed.Experiments show that the PIMNs are insensitive to boundary conditions and have excellent solution accuracy even with only the initial conditions.展开更多
基金Project supported by the National Key Research and Development Program of China(Grant No.2020YFC1807905)the National Natural Science Foundation of China(Grant Nos.52079090 and U20A20316)the Basic Research Program of Qinghai Province(Grant No.2022-ZJ-704).
文摘Neural network methods have been widely used in many fields of scientific research with the rapid increase of computing power.The physics-informed neural networks(PINNs)have received much attention as a major breakthrough in solving partial differential equations using neural networks.In this paper,a resampling technique based on the expansion-shrinkage point(ESP)selection strategy is developed to dynamically modify the distribution of training points in accordance with the performance of the neural networks.In this new approach both training sites with slight changes in residual values and training points with large residuals are taken into account.In order to make the distribution of training points more uniform,the concept of continuity is further introduced and incorporated.This method successfully addresses the issue that the neural network becomes ill or even crashes due to the extensive alteration of training point distribution.The effectiveness of the improved physics-informed neural networks with expansion-shrinkage resampling is demonstrated through a series of numerical experiments.
基金the funding from Anhui University of Science and Technology(No.2022yjrc15)the Key Project of National Natural Science Foundation of China(Nos.U21A20125 and U21A20122)+1 种基金the Key Research and Development Projects of Anhui Province(No.2022a05020043)the National Natural Science Foundation of China(Nos.51805410 and 51804007).
文摘In this work,a new method to solve the Reynolds equation including mass-conserving cavitation by using the physics informed neural networks(PINNs)is proposed.The complementarity relationship between the pressure and the void fraction is used.There are several difficulties in problem solving,and the solutions are provided.Firstly,the difficulty for considering the pressure inequality constraint by PINNs is solved by transferring it into one equality constraint without introducing error.While the void fraction inequality constraint is considered by using the hard constraint with the max-min function.Secondly,to avoid the fluctuation of the boundary value problems,the hard constraint method is also utilized to apply the boundary pressure values and the corresponding functions are provided.Lastly,for avoiding the trivial solution the limitation for the mean value of the void fraction is applied.The results are validated against existing data,and both the incompressible and compressible lubricant are considered.Good agreement can be found for both the domain and domain boundaries.
基金supported by the National Key Research and Development Program of China[grant number 2020YFA0608000]the National Natural Science Foundation of China[grant number 42075141]+2 种基金the Meteorological Joint Funds of the National Natural Science Foundation of China[grant number U2142211]the Key Project Fund of the Shanghai 2020“Science and Technology Innovation Action Plan”for Social Development[grant number 20dz1200702]the first batch of Model Interdisciplinary Joint Research Projects of Tongji University in 2021[grant number YB-21-202110].
基金the support from NSF DMS-1816783NVIDIA Corporation for their donation of a Quadro P6000 GPU for conducting some of the numerical simulations in this paper.
文摘In this paper,we introduce a new deep learning framework for discovering the phase-field models from existing image data.The new framework embraces the approximation power of physics informed neural networks(PINNs)and the computational efficiency of the pseudo-spectral methods,which we named pseudo-spectral PINN or SPINN.Unlike the baseline PINN,the pseudo-spectral PINN has several advantages.First of all,it requires less training data.A minimum of two temporal snapshots with uniform spatial resolution would be adequate.Secondly,it is computationally efficient,as the pseudo-spectral method is used for spatial discretization.Thirdly,it requires less trainable parameters compared with the baseline PINN,which significantly simplifies the training process and potentially assures fewer local minima or saddle points.We illustrate the effectiveness of pseudo-spectral PINN through several numerical examples.The newly proposed pseudo-spectral PINN is rather general,and it can be readily applied to discover other FDE-based models from image data.
文摘传统的数值求解方法面临维数灾难和效率与精度平衡问题,而基于数据驱动的神经网络求解方法又存在训练量冗余和不可解释性问题。针对此问题,物理信息神经网络(Physical Information Neural Networks,PINNs)关注了训练数据中隐含的物理先验知识,融合了神经网络拟合复杂变量的能力,赋予了传统神经网络所缺乏的物理可解释性。应用该算法模型,提出了一种基于PINN的Burgers方程求解模型,该算法模型在训练中施加物理信息约束,因此能用少量的训练样本学习预测到分布在时空域上的偏微分方程模型。实验结果表明,在1+1维Burgers方程算例下,所提方法相比于经典的机器学习算法能有效捕抓到方程的变化并进行精确模拟,相比于有限差分法,可以大幅度缩短模拟时间。通过对不同的网络参数进行比较实验,所提方法在10%的噪声破坏下能产生合理的识别准确度,网络逼近方程的待定系数误差在0.001以内。
文摘With the advent of physics informed neural networks(PINNs),deep learning has gained interest for solving nonlinear partial differential equations(PDEs)in recent years.In this paper,physics informed memory networks(PIMNs)are proposed as a new approach to solving PDEs by using physical laws and dynamic behavior of PDEs.Unlike the fully connected structure of the PINNs,the PIMNs construct the long-term dependence of the dynamics behavior with the help of the long short-term memory network.Meanwhile,the PDEs residuals are approximated using difference schemes in the form of convolution filter,which avoids information loss at the neighborhood of the sampling points.Finally,the performance of the PIMNs is assessed by solving the Kd V equation and the nonlinear Schr?dinger equation,and the effects of difference schemes,boundary conditions,network structure and mesh size on the solutions are discussed.Experiments show that the PIMNs are insensitive to boundary conditions and have excellent solution accuracy even with only the initial conditions.
