梯度更新不匹配：重新思考物理信息神经网络的无冲突训练

中文摘要 AI 辅助

训练物理信息神经网络（PINNs）需要联合优化物理残差损失项以及初始/边界条件损失项，这两类损失项往往会引发梯度冲突。梯度手术方法通过从特定损失的梯度中构建方向，在优化器变换前减少冲突，以此缓解该问题。然而，即便构建的方向无冲突，该特性在优化器变换后也可能无法保留。设$a_t$为梯度手术构建的方向，$u_t$为优化器的提议方向，$\boldsymbol{\textit{C}}_t$为特定损失梯度诱导的无冲突锥。我们发现，现代优化器会通过历史状态、自适应缩放、预条件或解耦权重衰减等机制变换$a_t$，因此$a_t \boldsymbol{\textit{C}}_t$通常并不意味着$u_t \boldsymbol{\textit{C}}_t$。我们将这种优化器诱导的、$a_t$与$u_t$之间无冲突性的差异称为梯度更新不匹配（GUM）。据此，我们提出梯度更新对齐（GUA）方法，该方法将$u_t$投影到$\boldsymbol{\textit{C}}_t$以获得对齐后的更新$p_t$，并将$p_t$应用于参数；当优化器维护内部状态时，GUA还会将该状态调整为从应用的更新中重构的目标状态。我们开展了大量实验，发现GUM在动量型、自适应型及基于曲率的优化器中普遍存在，冲突率最高达86.3%；在所有PINN设置中，GUA可实现无冲突的应用更新，并持续改进各类梯度手术方法，在单个设置中可将相对$L_2$误差降低多达98.2%。数据与代码可在该httpsURL获取。

英文摘要

Training Physics-Informed Neural Networks (PINNs) requires jointly optimizing physics residual and initial/boundary condition loss terms, which often induce conflicting gradients. Gradient surgery methods mitigate this issue by constructing directions from loss-specific gradients to reduce conflict before optimizer transformation. However, even when the constructed direction is conflict-free, this property may not be preserved after optimizer transformation. Let $a_t$ denote the direction constructed by gradient surgery, $u_t$ the optimizer proposal, and $\mathcal{C}_t$ the conflict-free cone induced by the loss-specific gradients. We show that modern optimizers can transform $a_t$ through mechanisms such as historical state, adaptive scaling, preconditioning, or decoupled weight decay, so $a_t \in \mathcal{C}_t$ does not generally imply $u_t \in \mathcal{C}_t$. We refer to this optimizer-induced discrepancy in conflict-freeness between $a_t$ and $u_t$ as Gradient-Update Mismatch (GUM). Accordingly, we propose Gradient-Update Alignment (GUA), which projects $u_t$ onto $\mathcal{C}_t$ to obtain the aligned update $p_t$ and applies $p_t$ to the parameters. When the optimizer maintains internal state, GUA further adjusts this state toward targets reconstructed from the applied update. We conduct extensive experiments and find that GUM is widespread across momentum, adaptive, and curvature-based optimizers, with conflict rates reaching up to 86.3%. Across all PINN settings, GUA achieves conflict-free applied updates and consistently improves various gradient surgery methods, reducing the relative $L_2$ error by up to 98.2% in individual settings. Data and code are available at https://github.com/JingXiao10/GUA.

发表机构

• National University of Defense Technology（国防科技大学）
• School of Computer Science and Technology, National University of Defense Technology（国防科技大学计算机学院）

机构由 AI 辅助整理，请以论文原文为准。