GMAdam：用任务级二阶矩调制改善 PINN 多任务损失不平衡问题
Task-Contribution Dynamics of Adam in Physics-Informed Neural Networks: Slow-Manifold Analysis and Gradient Modulation
训练 PINN 时总有一项损失压不住？这篇把 Adam 里各任务的贡献演化讲透了，还给出 GMAdam 这个即插即用的改法，实验里比 AdaTask 更准。
论文针对 PINN 中多个损失分量优化不平衡的问题，提出 Gradient-Modulated Adam（GMAdam）。作者先用任务级指标刻画 Adam 更新中各任务贡献的演化，识别出平滑变化、间歇尖峰和持续振荡三种训练阶段。通过慢流形一阶近似分析，论文指出任务贡献重分配由曲率对齐差异和预条件梯度分布差异驱动，两者都涉及 Adam 共享的二阶矩估计。GMAdam 借鉴 AdaTask，为每个任务使用独立的二阶矩估计来调制传入梯度。在高频 Poisson 方程、波动方程、Allen-Cahn 方程和二维顶盖驱动方腔流（Re=100,400）实验中，GMAdam 在相近计算成本下预测精度稳定优于 AdaTask。
Task-Contribution Dynamics of Adam in Physics-Informed Neural Networks: Slow-Manifold Analysis and Gradient Modulation
Physics-informed neural networks (PINNs) often suffer from imbalanced optimization of competing loss components, particularly when approximating solutions with high-frequency or multiscale features. Although existing balancing strategies have proven effective in alleviating this difficulty, the mechanisms governing the evolution of individual task contributions to Adam's parameter updates remain insufficiently understood. We introduce task-wise indicators to characterize this evolution and identify three distinct training regimes: smooth variation, intermittent spikes, and persistent oscillations. Focusing on the early smooth regime, we combine discrete-time analysis with a continuous-time singular perturbation framework under the full-batch setting and the parameter regime $(1-β_2)/(1-β_1)\ll1$. A first-order approximation of the locally attracting slow manifold reveals two factors governing the step-to-step redistribution of task contributions: curvature-related alignment disparities and differences in the coordinatewise distributions of preconditioned task gradients. Both factors involve Adam's shared second-moment estimate. Inspired by AdaTask, we propose Gradient-Modulated Adam (GMAdam), which modulates incoming task gradients using task-specific second-moment estimates. Numerical experiments on high-frequency Poisson and wave equations, the Allen-Cahn equation, and two-dimensional lid-driven cavity flow ($Re=100,400$) demonstrate consistent improvements in predictive accuracy over AdaTask at comparable computational cost.