提出可靠性感知的物理信息神经网络，提升复杂偏微分方程求解稳定性与精度。
Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations
- 引入可学习的可靠性调制场，动态调节内部表示，保持边界约束精确性
- 在多个挑战性问题上相对误差降低超60%，噪声与多模式场景下表现显著改善
- 适合处理局部剧烈变化、数据不可靠或多解结构的偏微分方程求解任务
物理信息神经网络（PINNs）为求解偏微分方程提供无网格框架，但常受损失不平衡、优化僵硬及难以捕捉局域或多模解结构的影响。硬-软PINNs（HSPINN）通过将Dirichlet或周期性约束嵌入试函数空间缓解部分困难，但固定可接受表示仍对尖锐或异质残差场条件不佳。本文提出可靠性感知的硬-软PINN（RA-HSPINN），在保持精确嵌入约束的同时，引入有界可学习的可靠性场以调节内部表示。该方法结合逆EMA全局损失平衡与轻量正则化，保留标准均方残差形式。可靠性场为数值调制变量，非物理参数或校准概率。在非线性Burgers方程、周期对流、混合边界泊松问题及一阶混合泊松系统上评估。相比HSPINN，RA-HSPINN在尖锐梯度Burgers中相对误差降低98.65%，噪声与不兼容初值条件下降低72.42%，光滑周期对流降低61.18%，局域周期对流降低60.02%，混合边界泊松降低29.36%，多模一阶泊松系统降低82.17%。结果表明，当硬-软试函数空间可行但难优化时，可靠性调制最有效，尤其在局域、不可靠数据和多模情形下。
原文摘要 · Abstract (English)
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.
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