提出新型神经网络框架,让物理方程求解更稳定高效。
Adaptive Hard-Soft Physics-Informed Neural Networks for Robust Boundary-Constrained PDE Solving

- 边界条件精确满足,其余作为软约束动态调整权重
- 在泊松、伯格斯等方程上收敛更快、精度更高
- 适合需高精度边界控制的科学计算场景
物理信息神经网络(PINNs)通过将物理规律嵌入学习过程来求解偏微分方程(PDE),但传统形式中所有约束均以软惩罚项形式加入复合损失函数,常导致收敛慢、对损失权重敏感且边界条件不准确。为此,本文提出统一的硬-软物理信息神经网络(HSPINN),采用自适应损失加权机制。其中,狄利克雷与周期性边界条件通过解析提升、掩码函数和周期特征映射精确强制执行,而控制PDE残差、诺伊曼通量及初始条件作为软约束处理。采用逆共享软最大策略动态平衡各损失成分,避免手动调参并提升梯度稳定性。该方法确保优化全程满足边界可容性,显著提高收敛效率与数值鲁棒性。在典型椭圆(泊松)、抛物(伯格斯)和双曲(带周期边界的对流)问题上的应用表明,HSPINN始终优于传统PINNs,在收敛速度、精度和稳定性方面表现更优,为跨科学与技术领域的物理约束深度学习提供了通用且可扩展的基础。
原文摘要 · Abstract (English)
Physics-informed neural networks (PINNs) provide an effective way to solve partial differential equations (PDEs) by embedding physical principles into the learning process. However, the conventional PINN formulation, in which all constraints are imposed as soft penalty terms within a composite loss, often exhibits slow convergence, sensitivity to loss weight scaling, and inaccurate boundary enforcement due to poor conditioning of the optimization landscape. To address these limitations, this study proposes a unified hard--soft physics--informed neural network (HSPINN) with adaptive loss weighting. In this framework, Dirichlet and periodic boundary conditions are enforced exactly by construction through analytical or polynomial lifting, masking functions, and periodic feature mappings, while the governing PDE residuals, Neumann fluxes, and initial conditions are treated as soft constraints. An inverse-share softmax strategy dynamically balances the relative importance of individual loss components during training, eliminating manual penalty tuning and improving gradient stability. This formulation ensures boundary admissibility throughout optimization and enhances convergence efficiency and numerical robustness. Applications to representative elliptic (Poisson), parabolic (Burgers), and hyperbolic (convection with periodic boundaries) problems demonstrate that HSPINN consistently achieves faster convergence, higher accuracy, and greater stability than conventional PINNs, establishing a general and scalable foundation for physics-constrained deep learning across science and technology.
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