A dynamic balanced physics-informed neural network for solving partial differential equations
In recent years,physics-informed neural networks(PINNs)have extensively been used in solving nonlinear partial differential equations(PDEs).PINNs add physical information as a regularization constraint to the neural network loss function,which can reduce the extensive reliance on data by traditional neural network methods.However,this approach makes PINN unable to dynamically adjust the individual residual weights in the loss function according to data changes during training,which leads to the limitation that PINN has large solution errors in solving nonlinear PDEs.In this paper,a Dynamic Balanced PINN(DBPINN)is proposed.Firstly,DBPINN designs a dynamic weight coefficient for each loss term of the PINN loss function,and uses a stochastic function to dynamically update the coefficient,which makes the PINN with only a single loss term converge better.Secondly,DBPINN establishes a balanced summation method for the loss terms of the PNNN loss function,which takes into account the competition among all the loss terms and leads to better convergence of the PINN as a whole.DBPINN makes PINN better optimized by dynamic weighting coefficients and a balanced summation method,which solves the problem of large solution error of PINN in practical applications.In this paper,four classical nonlinear PDEs in the field of scientific machine learning are selected for numerical validation and analysis of DBPINN.The experimental results show that DBPINN reduces the error by 46％and 64％than PINN on the Schrodinger and Allen-Cahn equations,respectively.The DBPINN reduces the error of coefficientλ1 by 1 to 2 orders of magnitude and the error of coefficient λ2 by about 50％in solving the Navier-Stokes equation,respectively.The DBPINN can reduce the error by 1 order of magnitude in multiple coefficients on the KdV equation.Finally,the performance and parameter ablation are verified on various forms of Burgers and Allen-Cahn equations,and the results show that DBPINN not only improves the model performance,handles small amounts of data,and fits equations in different time states,but also has better stability,accuracy,and convergence than PINN.DBPINN can be used instead of PINN for high accuracy solutions of various nonlinear PDEs.
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