用雅可比归一化提升物理神经网络求解强刚性方程的精度。
Solving stiff dark matter equations via Jacobian Normalization with Physics-Informed Neural Networks
- 通过雅可比矩阵归一化损失残差，缓解刚性微分方程的梯度问题。
- 在弱相互作用大质量粒子暗物质模型中实现更高精度，恢复完整解。
- 适用于暗物质反演问题，仅需一个观测数据点即可准确推断相互作用截面。
刚性微分方程严重制约物理信息神经网络（PINNs）的收敛性能。本文提出一种无需超参数、基于雅可比矩阵的损失残差归一化方法，理论上可改善梯度下降，并在经典刚性常微分方程基准上验证有效。进一步应用于真实物理系统——描述弱相互作用大质量粒子（WIMP）暗物质的刚性玻尔兹曼方程（BEs），该方法在精度上超越此前采用注意力机制的方法，成功恢复此前方法失效的完整解。在仅含一个实验数据点（即观测到的暗物质遗迹密度）的反演问题中，我们的反演PINNs在标准及替代宇宙学框架下均能正确推断出满足玻尔兹曼方程的相互作用截面。
原文摘要 · Abstract (English)
Stiff differential equations pose a major challenge for Physics-Informed Neural Networks (PINNs), often causing poor convergence. We propose a simple, hyperparameter-free method to address stiffness by normalizing loss residuals with the Jacobian. We provide theoretical indications that Jacobian-based normalization can improve gradient descent and validate it on benchmark stiff ordinary differential equations. We then apply it to a realistic system: the stiff Boltzmann equations (BEs) governing weakly interacting massive particle (WIMP) dark matter (DM). Our approach achieves higher accuracy than attention mechanisms previously proposed for handling stiffness, recovering the full solution where prior methods fail. This is further demonstrated in an inverse problem with a single experimental data point - the observed DM relic density - where our inverse PINNs correctly infer the cross section that solves the BEs in both Standard and alternative cosmologies.
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