发表机构
College of Engineering and Computer Science, VinUniversity; Center for AI Research, VinUniversity; Department of Computer Science and Engineering, University of North Texas; Department of Mathematics and Informatics, Hanoi University of Science and Technology; Group of Materials and Structures, School of Mechanical Engineering, Hanoi University of Science and Technology（越南 Vin 大学工程与计算机科学学院; 越南 Vin 大学人工智能研究中心; 北德克萨斯大学计算机科学与工程系; 河内科学技术大学数学与信息学系; 河内科学技术大学机械工程学院材料与结构组）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究针对物理信息神经网络训练难题，提出可靠性感知硬软PINN（RA - HSPINN），结合可靠性感知假设、逆指数移动平均全局损失平衡和轻量级正则化，在多个偏微分方程问题上评估，相比HSPINN大幅降低相对误差，凸显该方法优势。
AI 中文摘要
物理信息神经网络（PINNs）为求解偏微分方程提供了无网格框架，但其训练常受损失不平衡、优化刚度以及捕获局部或多模式解结构困难的影响。硬软PINNs（HSPINN）通过将狄利克雷或周期约束直接嵌入试验空间缓解了部分困难，但所得固定的可允许表示对于尖锐或异质残差场仍可能条件不佳。本文提出了一种可靠性感知硬软PINN（RA - HSPINN），它在引入有界可学习可靠性场来调制内部表示的同时保留精确的嵌入约束。该方法将这种可靠性感知假设与逆指数移动平均全局损失平衡和轻量级正则化相结合，同时保留标准均方残差形式。可靠性场是一个数值调制变量，而非物理参数或校准概率。在非线性伯格斯方程、周期对流、混合边界泊松问题和混合一阶泊松系统上对RA - HSPINN进行了评估。与HSPINN相比，对于尖锐梯度伯格斯方程相对误差降低了98.65%，对于带有噪声和不兼容初始条件的伯格斯数据降低了72.42%，对于平滑周期对流降低了61.18%，对于局部周期对流降低了60.02%，对于混合边界泊松问题降低了29.36%，对于多模式混合一阶泊松系统降低了82.17%。结果表明，当硬软试验空间可允许但难以优化时，尤其是在局部、不可靠数据和多模式偏微分方程 regime中，可靠性感知调制最为有益。
英文摘要
Physics-informed neural networks (PINNs) provide a mesh-free framework for solving partial differential equations, but their training is often affected by loss imbalance, optimization stiffness, and difficulty in capturing localized or multi-mode solution structures. Hard-soft PINNs (HSPINN) alleviate part of this difficulty by embedding Dirichlet or periodic constraints directly into the trial space, but the resulting fixed admissible representation can still be poorly conditioned for sharp or heterogeneous residual fields. This paper proposes a reliability-aware hard-soft PINN (RA-HSPINN) that preserves exact embedded constraints while introducing a bounded learnable reliability field to modulate the interior representation. The method combines this reliability-aware ansatz with inverse-EMA global loss balancing and lightweight regularization, while retaining the standard mean-square residual form. The reliability field is a numerical modulation variable, not a physical parameter or calibrated probability. RA-HSPINN is evaluated on nonlinear Burgers equations, periodic convection, a mixed-boundary Poisson problem, and a mixed first-order Poisson system. Compared with HSPINN, it reduces the relative error by $98.65%$ for sharp-gradient Burgers, $72.42%$ for Burgers data with noisy and incompatible initial conditions, $61.18%$ for smooth periodic convection, $60.02%$ for localized periodic convection, $29.36%$ for mixed-boundary Poisson, and $82.17%$ for a multi-mode mixed first-order Poisson system. The results show that reliability-aware modulation is most beneficial when hard-soft trial spaces are admissible but difficult to optimize, especially in localized, unreliable-data, and multi-mode PDE regimes.
