Curriculum-Enhanced Adaptive Sampling for Physics-Informed Neural Networks: A Robust Framework for Stiff PDEs

Hasan Cetinkaya, Fahrettin Ay, Mehmet Tunçel, Hazem Nounou, Mohamed Numan Nounou, Hasan Kurban, Erchin Serpedin

Abstract: Physics-Informed Neural Networks (PINNs) often struggle with stiff partial differential equations (PDEs) exhibiting sharp gradients and extreme nonlinearities. We propose a Curriculum-Enhanced (CE) Adaptive Sampling framework that integrates curriculum learning with adaptive refinement to improve PINN training. Our framework introduces four methods: CE-RARG (greedy sampling), CE-RARD (probabilistic sampling), and their novel difficulty-aware dynamic counterparts, CED-RARG and CED-RARD, which adjust refinement effort based on task difficulty. We test these methods on five challenging stiff PDEs: the Allen–Cahn, Burgers’ (I and II), Korteweg–de Vries (KdV), and Reaction equations. Our methods consistently outperform both Vanilla PINNs and curriculum-only baselines. In the most difficult regimes, CED-RARD achieves errors up to 100 times lower for the Burgers’ and KdV equations. For the Allen–Cahn and Reaction equations, CED-RARG proves most effective, reducing errors by over 40% compared to its non-dynamic counterpart and by over two orders of magnitude relative to Vanilla PINN. Visualizations confirm that our methods effectively allocate collocation points to high-gradient regions. By demonstrating success across a wide range of stiffness parameters, we provide a robust and reproducible framework for solving stiff PDEs, with all code and datasets publicly available.

External IDs:doi:10.3390/math13243996