arXiv:2601.12971cs.LG2026-01被引 2

通过注意力机制与梯度冲突缓解,提升物理信息神经网络的求解精度与稳定性。

Architecture-Optimization Co-Design for Physics-Informed Neural Networks Via Attentive Representations and Conflict-Resolved Gradients

  • 引入分层动态注意力增强模型表达能力,解决表征受限问题。
  • 提出冲突化解梯度更新策略,降低多物理约束间的梯度干扰。
  • 在多个经典偏微分方程上实现更快收敛和更低误差,适合高精度科学计算场景。

物理信息神经网络(PINNs)通过将物理定律嵌入训练过程,为求解偏微分方程(PDEs)提供了一种基于学习的框架。然而,其性能常受制于有限的表征能力及由竞争性物理约束引发的优化困难。本文从架构-优化协同设计视角出发,提出分层动态注意力机制以增强表征灵活性,构建了LDA-PINN;同时将PINN训练重构为多任务学习问题,引入冲突化解梯度更新策略,形成GC-PINN。二者融合后得到ACR-PINN,结合注意力表示与冲突感知优化,保持标准PINN损失形式。在Burgers、Helmholtz、Klein-Gordon及驱动腔流等基准PDE问题上的大量实验表明,ACR-PINN相比标准PINN实现了更快收敛以及显著更低的相对 $L_2$ 与 $L_ ty$ 误差,验证了架构-优化协同设计在提升PINN求解器鲁棒性与精度方面的有效性。

原文摘要 · Abstract (English)

Physics-Informed Neural Networks (PINNs) provide a learning-based framework for solving partial differential equations (PDEs) by embedding governing physical laws into neural network training. In practice, however, their performance is often hindered by limited representational capacity and optimization difficulties caused by competing physical constraints and conflicting gradients. In this work, we study PINN training from a unified architecture-optimization perspective. We first propose a layer-wise dynamic attention mechanism to enhance representational flexibility, resulting in the Layer-wise Dynamic Attention PINN (LDA-PINN). We then reformulate PINN training as a multi-task learning problem and introduce a conflict-resolved gradient update strategy to alleviate gradient interference, leading to the Gradient-Conflict-Resolved PINN (GC-PINN). By integrating these two components, we develop the Architecture-Conflict-Resolved PINN (ACR-PINN), which combines attentive representations with conflict-aware optimization while preserving the standard PINN loss formulation. Extensive experiments on benchmark PDEs, including the Burgers, Helmholtz, Klein-Gordon, and lid-driven cavity flow problems, demonstrate that ACR-PINN achieves faster convergence and significantly lower relative $L_2$ and $L_\infty$ errors than standard PINNs. These results highlight the effectiveness of architecture-optimization co-design for improving the robustness and accuracy of PINN-based solvers.

PINN偏微分方程神经网络优化注意力机制

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