Title: Adaptive Spatial-Temporal Generalization for Physics-Informed Neural PDE Solvers
Primary area: learning on time series and dynamical systems
Keywords: Physics-Informed Neural Networks (PINNs), Neural PDE solvers, Adaptive weighting, Generalization, Convergence acceleration

Abstract: Integrating domain knowledge into neural networks has advanced the development of Physics-Informed Neural Networks (PINNs), enabling solutions to partial differential equations across diverse applications. To enhance generalization in neural PDE solvers, we propose MagniLearning, an adaptive weighting strategy that dynamically adjusts the importance of spatial regions, knowledge components, and temporal segments during training. Our approach evaluates the impact of omitting each region or time block on model performance and assigns higher weights to the most influential data. This adaptive scheme accelerates convergence in neural PDE solvers by emphasizing the most informative regions and time segments, while enhancing robustness to noise and underrepresented physics. We formalize the method using an effective risk function that incorporates region- and time-dependent weights, and we provide theoretical guarantees for controlling the generalization error. Numerical experiments demonstrate that MagniLearning significantly improves both stability and accuracy.