通过交替训练提升PINN求解PDE的精度与稳定性
More Consistent Accuracy PINN via Alternating Easy-Hard Training
- 交替使用难易样本优先策略，融合两种方法优势
- 在复杂PDE上相对L2误差达10^-5至10^-6量级
- 适合需要高可靠性的科学计算场景
物理信息神经网络（PINNs）近年来成为求解偏微分方程（PDEs）的重要方法，但其训练策略仍不充分。尽管受有限元方法启发的难样本优先法被广泛采用，近期研究也表明易样本优先同样有效，但我们发现两者均存在明显权衡且在不同PDE类型上表现不一致。为此，我们提出一种混合策略，通过交替训练算法结合难易优先的优势。在具有陡峭梯度、非线性和高维度的PDE上，该方法实现了一致的高精度，相对L2误差大多处于10^-5至10^-6量级，显著优于基线方法。此外，其在多种问题上表现出更强的可靠性，而对比方法则常因PDE类型不同导致精度波动。本工作为设计增强PINNs性能与鲁棒性的混合训练策略提供了新思路。
原文摘要 · Abstract (English)
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.
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