An Energy-Aware Sampling Method for Physics-Informed Neural Networks

Adaptive sampling is a critical technique in Physics-Informed Neural Networks (PINNs), particularly for solving stiff and multiscale partial differential equations (PDEs). Existing adaptive sampling methods predominantly rely on the PDE residual. However, residuals measure local equation violations rather than true approximation errors, frequently neglecting regions with large actual errors, which leads to inefficient resource allocation and slow convergence. To overcome this limitation, we explore an energy-aware sampling paradigm that leverages the intrinsic structure of energy-driven PDEs. Specifically, focusing on phase-field equations as a representative class, our theoretical analysis reveals that the double-well potential not only identifies physically important regions, but also dictates the necessity of a higher collocation point density in high-potential regions to achieve uniform generalization accuracy. Motivated by these insights, we propose Potential-based Adaptive Sampling (PAS), which exploits the derivative-free nature of the potential to integrate global exploration with local refinement. Extensive comparisons demonstrate that PAS significantly outperforms residual-based baselines in accuracy, convergence speed, and training efficiency.