提出正则化方法，让物理神经网络避开不稳定平衡点，提升求解准确率。
Stabilizing PINNs: A regularization scheme for PINN training to avoid unstable fixed points of dynamical systems
- 基于稳定性理论，对不稳定平衡点施加惩罚项
- 在4个动力系统上显著提高训练成功率，避免错误解
- 适合需要高精度求解微分方程的科研与工程场景
近期研究表明，用于训练物理信息神经网络（PINNs）的损失函数在动力系统的平衡点处存在局部极小值。在前向求解设置中，即训练PINN求解初值问题时，这些局部极小值可能干扰训练过程，导致物理上不正确的解。本文基于稳定性理论，提出一种正则化方案，通过惩罚对应于不稳定平衡点的解来改善训练。在四个动力系统上的实验结果表明，该方法有助于避免物理上不正确的解，并显著提升PINNs的训练成功率，包括洛特卡-沃尔泰拉模型和范德波尔振子。
原文摘要 · Abstract (English)
It was recently shown that the loss function used for training physics-informed neural networks (PINNs) exhibits local minima at solutions corresponding to fixed points of dynamical systems. In the forward setting, where the PINN is trained to solve initial value problems, these local minima can interfere with training and potentially leading to physically incorrect solutions. Building on stability theory, this paper proposes a regularization scheme that penalizes solutions corresponding to unstable fixed points. Experimental results on four dynamical systems, including the Lotka-Volterra model and the van der Pol oscillator, show that our scheme helps avoiding physically incorrect solutions and substantially improves the training success rate of PINNs.
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