More Consistent Accuracy PINN via Alternating Easy-Hard Training

TL;DR

This work addresses the inconsistent performance of PINNs arising from imbalanced loss components by introducing AEH-PINN, an alternating hybrid training scheme that cycles between hard-prioritization (min–max weighted loss) and easy-prioritization (progressive easy-sample selection). Across six challenging PDEs, AEH-PINN achieves consistently superior relative $L^2$ errors, with notable gains in steep-gradient, nonlinear, and high-dimensional multiscale problems, and ablation confirms the benefit of the alternating mechanism. The study demonstrates faster and more robust convergence dynamics while maintaining comparable model size and computational cost, offering a practical, architecture-agnostic improvement to PINN training. These results suggest that hybrid, curriculum-like sampling strategies can significantly enhance the reliability and accuracy of physics-informed learning in complex PDE regimes.

Abstract

Physics-informed neural networks (PINNs) have recently emerged as a prominent paradigm for solving partial differential equations (PDEs), yet their training strategies remain underexplored. While hard prioritization methods inspired by finite element methods are widely adopted, recent research suggests that easy prioritization can also be effective. Nevertheless, we find that both approaches exhibit notable trade-offs and inconsistent performance across PDE types. To address this issue, we develop a hybrid strategy that combines the strengths of hard and easy prioritization through an alternating training algorithm. On PDEs with steep gradients, nonlinearity, and high dimensionality, the proposed method achieves consistently high accuracy, with relative L2 errors mostly in the range of O(10^-5) to O(10^-6), significantly surpassing baseline methods. Moreover, it offers greater reliability across diverse problems, whereas compared approaches often suffer from variable accuracy depending on the PDE. This work provides new insights into designing hybrid training strategies to enhance the performance and robustness of PINNs.

Figures (21)

• Figure 1: Visualization of a source term at $\alpha=0.11$ used in the heat conduction equation. Left: 3D surface of the source term $f(x,t)$, exhibiting sharp localized peaks and steep gradients, with value ranges exceeding $10^5$. Right: 1D slice of $f(x,t)$ along $x = 0$, showing highly nontrivial temporal behavior. Such source terms introduce strong local features and multiscale variations in the solution, posing significant challenges for standard PINNs to learn effectively.
• Figure 2: Relative $L^2$ errors of SAPINN and AAPINN on equation \ref{['toy']} with different difficulty levels ($\alpha = 0.3, 0.15, 0.11$).
• Figure 3: Dynamics of weight and error distribution in SAPINN for equation ($\alpha=0.11$) at different training stages.
• Figure 4: Absolute error distribution comparison of different methods for the heat conduction problem \ref{['Heat']}.
• Figure 5: Absolute error distribution comparison of different methods for the Helmholtz equation \ref{['Helm']}.