FAMAW-PINN: A physics-informed neural network integrating adaptive loss weighting with firefly-inspired adaptive point movement

The Physics-Informed Neural Network (PINN) uses automatic differentiation to embed the governing partial differential equation (PDE) into the neural network’s loss function as physical constraints, providing a powerful approach for solving forward and inverse PDE problems. During training, physical loss calculation relies on predefined spatiotemporal collocation points. However, when solving equations with steep gradients or singularities, conventional fixed or randomly distributed training points often fail to capture critical solution structures, reducing PINN’s prediction accuracy. Inspired by firefly phototaxis, this paper proposes a bio-inspired dynamic training point movement strategy named firefly adaptive collocation point movement (FAM). Its core mechanism uses the neural network’s residual or gradient as the "brightness" signal (analogous to firefly perception) to drive training points toward high-residual or high-gradient regions in the computational domain, thereby capturing key physical features. To further enhance PINN’s performance, we integrate FAM with an adaptive loss weighting technique (AW), forming a new adaptive strategy termed FAMAW. This strategy dynamically balances the migration of training points and the weighting of loss terms. Numerical experiments and comparisons with established methods demonstrate the significant superiority of FAMAW-PINN and its marked improvement in solution accuracy.