The solution of fractional partial differential equations(PDEs)is an important topic in scientific computing.However,the traditional physics-informed neural networks(PINNs)have problems of memory overflow and low comp...The solution of fractional partial differential equations(PDEs)is an important topic in scientific computing.However,the traditional physics-informed neural networks(PINNs)have problems of memory overflow and low computational efficiency when the derivative is discretized for a long time.Therefore in this paper we innovatively propose a framework of Laplace transform physics-informed neural networks(LT-PINNs),which is dedicated to solving the forward and inverse problems of Caputo-type fractional PDEs.The core of this method is to use the Laplace transform to construct the loss function,which skillfully avoids the dilemma that the fractional derivative operator in traditional PINNs is difficult to operate effectively.By studying the benchmark problem of parameter a in a series of different scenarios we verify that LT-PINNs can predict the solution of Caputo-type fractional PDEs more accurately than fractional PINNs.The excellent performance of LT-PINNs in identifying inverse problems involving fractional order,convection and diffusion coefficients is further explored.At the same time,the effects of network structure,the number of sampling points and noise on the LT-PINNs method are analyzed in detail.The results show that the method can predict the solution of the equation satisfactorily even under severe noise interference.The proposed LT-PINNs framework opens up a new path for efficiently solving fractional PDEs.It shows significant advantages in improving computational efficiency,reducing memory usage and dealing with complex noise environments.It is expected to promote the further development of fractional PDEs in many fields.展开更多
The classical heat conduction equation is derived from the assumption that the temperature increases immediately after heat transfer, but the increase of temperature is a slow process, so the memory-dependent heat con...The classical heat conduction equation is derived from the assumption that the temperature increases immediately after heat transfer, but the increase of temperature is a slow process, so the memory-dependent heat conduction model has been reconstructed. Numerical results show that the solution of the initial boundary value problem of the new model is similar to that of the classical heat conduction equation, but its propagation speed is slower than that of the latter. In addition, the propagation speed of the former is also affected by time delay and kernel function.展开更多
基金funded by the National Natural Science Foundation of China(Grant No.12061055)the Key Projects of the Natural Science Foundation of Ningxia Hui Autonomous Region of China(Grant No.2022AAC02005)the Scientific and Technological Innovation Leading Talent Project of Ningxia Hui Autonomous Region of China(Grant No.2021GKLRLX06)。
文摘The solution of fractional partial differential equations(PDEs)is an important topic in scientific computing.However,the traditional physics-informed neural networks(PINNs)have problems of memory overflow and low computational efficiency when the derivative is discretized for a long time.Therefore in this paper we innovatively propose a framework of Laplace transform physics-informed neural networks(LT-PINNs),which is dedicated to solving the forward and inverse problems of Caputo-type fractional PDEs.The core of this method is to use the Laplace transform to construct the loss function,which skillfully avoids the dilemma that the fractional derivative operator in traditional PINNs is difficult to operate effectively.By studying the benchmark problem of parameter a in a series of different scenarios we verify that LT-PINNs can predict the solution of Caputo-type fractional PDEs more accurately than fractional PINNs.The excellent performance of LT-PINNs in identifying inverse problems involving fractional order,convection and diffusion coefficients is further explored.At the same time,the effects of network structure,the number of sampling points and noise on the LT-PINNs method are analyzed in detail.The results show that the method can predict the solution of the equation satisfactorily even under severe noise interference.The proposed LT-PINNs framework opens up a new path for efficiently solving fractional PDEs.It shows significant advantages in improving computational efficiency,reducing memory usage and dealing with complex noise environments.It is expected to promote the further development of fractional PDEs in many fields.
文摘The classical heat conduction equation is derived from the assumption that the temperature increases immediately after heat transfer, but the increase of temperature is a slow process, so the memory-dependent heat conduction model has been reconstructed. Numerical results show that the solution of the initial boundary value problem of the new model is similar to that of the classical heat conduction equation, but its propagation speed is slower than that of the latter. In addition, the propagation speed of the former is also affected by time delay and kernel function.
