Predictive Moving Sample Method for Physics-Informed Neural Solvers of Time-Dependent PDEs

Time-dependent partial differential equations (PDEs) often develop sharp fronts, localized peaks, and other moving structures that occupy only a small portion of the space--time domain but dominate the approximation error. This makes fixed or uniformly sampled collocation strategies inefficient for physics-informed neural networks (PINNs), especially in high dimensions and over long-time prediction intervals. We propose the predictive moving sample method (PMSM), which builds on the moving sample method (MSM) in \cite{xu2026moving} by replacing its full time domain iterative training with a progressive time-stepping strategy and simplifying the velocity-field loss to further reduce the per-step cost. To improve practicality for long-time prediction, we further introduce the windowed-reset predictive moving sample method (WR-PMSM), which restricts extension training to an active time window and periodically resets the reference state, thereby reducing the growth of optimization cost while preserving global consistency through a final refinement stage. Across four representative benchmarks, PMSM consistently outperforms both standard PINNs and the original MSM under matched collocation budgets. These results suggest that transporting samples according to residual dynamics provides an effective and practical route to neural network solvers for time-dependent PDEs.