Warm-Up-Calibrated Adaptive Weighting for Spatiotemporal Residual-Derivative Balance in Gradient-Enhanced PINNs

Neural-network methods are widely used for partial differential equations (PDEs). Within this framework, gradient-enhanced physics-informed neural networks (gPINNs) incorporate first-order residual derivatives, while fixed directional weighting may unbalance spatial and temporal residual-gradient contributions. Accordingly, an adaptive gradient-balanced gPINN (AGB-gPINN) is proposed using warm-up calibration, dimensionless relative states, and a bounded fixed-sum directional budget without trainable weighting parameters. For Burgers, median relative L 2 error and PDE-MSE over nine seeds decrease by 41.9% and 57.0% relative to fixed-weight gPINN; Allen–Cahn shows smaller gains and lower variability. These results support more balanced and stable gPINN training.