用轻量几何感知优化，提升物理神经网络的收敛与稳定性。
Lightweight Geometric Adaptation for Training Physics-Informed Neural Networks
- 通过梯度差估计局部几何变化，自适应调整优化步长。
- 在多个复杂方程上实现更快收敛、更稳定训练和更高精度。
- 无需计算二阶矩阵，可无缝集成现有优化器，适合高维问题。
物理信息神经网络（PINNs）常因损失函数景观的各向异性和快速变化几何导致收敛慢、训练不稳定及精度下降。本文提出一种轻量级曲率感知优化框架，通过连续梯度差作为局部几何变化的廉价代理，结合步长归一化的割线曲率指标来控制校正强度，为现有的一阶优化器添加自适应预测校正机制。该框架即插即用、计算高效，无需显式构建二阶矩阵。在多种偏微分方程基准测试中，包括高维热方程、Gray--Scott系统、Belousov--Zhabotinsky系统和二维Kuramoto--Sivashinsky系统，均显著优于标准优化器和强基线，表现为收敛速度加快、训练更稳定、解的精度更高。
原文摘要 · Abstract (English)
Physics-Informed Neural Networks (PINNs) often suffer from slow convergence, training instability, and reduced accuracy on challenging partial differential equations due to the anisotropic and rapidly varying geometry of their loss landscapes. We propose a lightweight curvature-aware optimization framework that augments existing first-order optimizers with an adaptive predictive correction based on secant information. Consecutive gradient differences are used as a cheap proxy for local geometric change, together with a step-normalized secant curvature indicator to control the correction strength. The framework is plug-and-play, computationally efficient, and broadly compatible with existing optimizers, without explicitly forming second-order matrices. Experiments on diverse PDE benchmarks show consistent improvements in convergence speed, training stability, and solution accuracy over standard optimizers and strong baselines, including on the high-dimensional heat equation, Gray--Scott system, Belousov--Zhabotinsky system, and 2D Kuramoto--Sivashinsky system.
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