改进物理信息神经网络优化，实现机器精度求解微分方程
AMStraMGRAM: Adaptive Multi-cutoff Strategy Modification for ANaGRAM
- 引入多截断自适应策略，动态调整奇异值分解正则化强度
- 在标准偏微分方程测试中达到机器精度，收敛速度显著提升
- 理论证明正则化必要性，并连接格林函数与谱分析
近期研究显示，自然梯度方法在训练物理信息神经网络（PINNs）时可显著优于传统优化器。本文分析了基于奇异值分解与截断正则化的自然梯度方法ANaGRAM的训练动态，提出一种多截断自适应策略以进一步提升其性能。在基准偏微分方程上的实验验证了该方法的有效性，部分任务可达到机器精度。为提供理论支撑，我们构建了基于谱理论的分析框架，解释了正则化的必要性，并拓展了与格林函数理论的已有联系。
原文摘要 · Abstract (English)
Recent works have shown that natural gradient methods can significantly outperform standard optimizers when training physics-informed neural networks (PINNs). In this paper, we analyze the training dynamics of PINNs optimized with ANaGRAM, a natural-gradient-inspired approach employing singular value decomposition with cutoff regularization. Building on this analysis, we propose a multi-cutoff adaptation strategy that further enhances ANaGRAM's performance. Experiments on benchmark PDEs validate the effectiveness of our method, which allows to reach machine precision on some experiments. To provide theoretical grounding, we develop a framework based on spectral theory that explains the necessity of regularization and extend previous shown connections with Green's functions theory.
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