Physics-informed neural networks(PINNs)have been shown as powerful tools for solving partial differential equations(PDEs)by embedding physical laws into the network training.Despite their remarkable results,complicate...Physics-informed neural networks(PINNs)have been shown as powerful tools for solving partial differential equations(PDEs)by embedding physical laws into the network training.Despite their remarkable results,complicated problems such as irregular boundary conditions(BCs)and discontinuous or high-frequency behaviors remain persistent challenges for PINNs.For these reasons,we propose a novel two-phase framework,where a neural network is first trained to represent shape functions that can capture the irregularity of BCs in the first phase,and then these neural network-based shape functions are used to construct boundary shape functions(BSFs)that exactly satisfy both essential and natural BCs in PINNs in the second phase.This scheme is integrated into both the strong-form and energy PINN approaches,thereby improving the quality of solution prediction in the cases of irregular BCs.In addition,this study examines the benefits and limitations of these approaches in handling discontinuous and high-frequency problems.Overall,our method offers a unified and flexible solution framework that addresses key limitations of existing PINN methods with higher accuracy and stability for general PDE problems in solid mechanics.展开更多
Enforcing initial and boundary conditions(I/BCs)poses challenges in physics-informed neural networks(PINNs).Several PINN studies have gained significant achievements in developing techniques for imposing BCs in static...Enforcing initial and boundary conditions(I/BCs)poses challenges in physics-informed neural networks(PINNs).Several PINN studies have gained significant achievements in developing techniques for imposing BCs in static problems;however,the simultaneous enforcement of I/BCs in dynamic problems remains challenging.To overcome this limitation,a novel approach called decoupled physics-informed neural network(d PINN)is proposed in this work.The d PINN operates based on the core idea of converting a partial differential equation(PDE)to a system of ordinary differential equations(ODEs)via the space-time decoupled formulation.To this end,the latent solution is expressed in the form of a linear combination of approximation functions and coefficients,where approximation functions are admissible and coefficients are unknowns of time that must be solved.Subsequently,the system of ODEs is obtained by implementing the weighted-residual form of the original PDE over the spatial domain.A multi-network structure is used to parameterize the set of coefficient functions,and the loss function of d PINN is established based on minimizing the residuals of the gained ODEs.In this scheme,the decoupled formulation leads to the independent handling of I/BCs.Accordingly,the BCs are automatically satisfied based on suitable selections of admissible functions.Meanwhile,the original ICs are replaced by the Galerkin form of the ICs concerning unknown coefficients,and the neural network(NN)outputs are modified to satisfy the gained ICs.Several benchmark problems involving different types of PDEs and I/BCs are used to demonstrate the superior performance of d PINN compared with regular PINN in terms of solution accuracy and computational cost.展开更多
基金Project supported by the Basic Science Research Program through the National Research Foundation(NRF)of Korea funded by the Ministry of Science and ICT(No.RS-2024-00337001)。
文摘Physics-informed neural networks(PINNs)have been shown as powerful tools for solving partial differential equations(PDEs)by embedding physical laws into the network training.Despite their remarkable results,complicated problems such as irregular boundary conditions(BCs)and discontinuous or high-frequency behaviors remain persistent challenges for PINNs.For these reasons,we propose a novel two-phase framework,where a neural network is first trained to represent shape functions that can capture the irregularity of BCs in the first phase,and then these neural network-based shape functions are used to construct boundary shape functions(BSFs)that exactly satisfy both essential and natural BCs in PINNs in the second phase.This scheme is integrated into both the strong-form and energy PINN approaches,thereby improving the quality of solution prediction in the cases of irregular BCs.In addition,this study examines the benefits and limitations of these approaches in handling discontinuous and high-frequency problems.Overall,our method offers a unified and flexible solution framework that addresses key limitations of existing PINN methods with higher accuracy and stability for general PDE problems in solid mechanics.
基金Project supported by the Basic Science Research Program through the National Research Foundation(NRF)of Korea funded by the Ministry of Science and ICT(No.RS-2024-00337001)。
文摘Enforcing initial and boundary conditions(I/BCs)poses challenges in physics-informed neural networks(PINNs).Several PINN studies have gained significant achievements in developing techniques for imposing BCs in static problems;however,the simultaneous enforcement of I/BCs in dynamic problems remains challenging.To overcome this limitation,a novel approach called decoupled physics-informed neural network(d PINN)is proposed in this work.The d PINN operates based on the core idea of converting a partial differential equation(PDE)to a system of ordinary differential equations(ODEs)via the space-time decoupled formulation.To this end,the latent solution is expressed in the form of a linear combination of approximation functions and coefficients,where approximation functions are admissible and coefficients are unknowns of time that must be solved.Subsequently,the system of ODEs is obtained by implementing the weighted-residual form of the original PDE over the spatial domain.A multi-network structure is used to parameterize the set of coefficient functions,and the loss function of d PINN is established based on minimizing the residuals of the gained ODEs.In this scheme,the decoupled formulation leads to the independent handling of I/BCs.Accordingly,the BCs are automatically satisfied based on suitable selections of admissible functions.Meanwhile,the original ICs are replaced by the Galerkin form of the ICs concerning unknown coefficients,and the neural network(NN)outputs are modified to satisfy the gained ICs.Several benchmark problems involving different types of PDEs and I/BCs are used to demonstrate the superior performance of d PINN compared with regular PINN in terms of solution accuracy and computational cost.
