摘要
This paper presents ALM-PINN,an adaptive physical informed neural network algorithm optimized by Levenberg-Marquardt.ALM-PINN is tailored to overcome challenges for solving singular perturbation problems(SPP).Traditional neural network-based solvers reframe solving differential equations task as a multi-objective optimization problem involving residual or Ritz error.However,significant disparities in the magnitudes of loss functions and their gradients frequency result in suboptimal training and convergence challenges.Addressing these issues,ALM-PINN introduces a learnable parameter for the perturbation parameter and constructs a two-terms loss function.The first loss term emphasizes approximating the governing equation,while the second term minimizes the difference between perturbation and learnable parameters.This adaptive learning strategy not only mitigates convergence issues in directly solving SPP but also alleviates the computational burden with asymptotic iteration from a large initial value.For one-dimensional tasks,ALM-PINN enhances training efficiency and reduces complexity by enforcing hard constraints on boundary conditions,streamlining the loss function sub-terms.The efficacy of ALM-PINN is evaluated on five SPPs,demonstrating its ability to capture sharp changes in physical quantities within the boundary layer,even with small perturbation coefficients.Furthermore,ALM-PINN exhibits reduced errors in both L¥and L2 norms,coupled with improved convergence speed and stability.
基金
supported by the Fundamental Research Funds from the Central South University(Grant No.2022zyts0611)
the China Scholarship Council(Grant No.202206370078)
supported by the Natural Science Foundation of Hunan Province(Grant Nos.2023JJ30648 and 2022JJ40461)
the Natural Science Foundation of Changsha(Grant No.kq2208252)
by the Excellent Youth Foundation of Education Bureau of Hunan Province(Grant No.21B0301).
