AI 中文总结
研究针对物理信息神经网络同步优化与物理传播路径不符问题，基于物理信息传播路径定义训练优先级，用NTK动力学分析原因，提出统一多维优先级约束框架，改善了PINNs在特定问题上的收敛行为和预测准确性。
AI 中文摘要
物理信息神经网络（PINNs）在求解偏微分方程方面显示出潜力，但同步优化对不同区域和约束的残差一视同仁，与“从源到响应”的物理信息传播路径不符，降低训练稳定性和准确性。现有因果训练方法主要关注时间维度，缺乏对空间和边界维度的统一表征。为此，我们根据物理信息传播路径定义了统一的训练优先级类别，利用神经切线核（NTK）动力学理论分析了标准PINNs不遵循此优先级的原因，提出了统一的多维优先级约束框架，通过划分域并构建负指数残差权重，将物理传播顺序转化为训练优先级。基准案例表明，该方法在不修改网络架构且计算成本可控的情况下，持续改善了PINNs在具有明确传播路径或约束主导结构问题上的收敛行为和预测准确性。
英文摘要
Physics-informed neural networks (PINNs) have shown promise for solving partial differential equations (PDEs); however, their synchronous optimization treats residuals of different regions and constraints equally, which is inconsistent with the progressive "from source to response" physical information propagation path, degrading training stability and accuracy. Existing causal training methods focus mainly on the temporal dimension, lacking a unified characterization of spatial and boundary dimensions. To address this, we define a unified class of training priorities according to the physical information propagation path: premise regions should be learned before dependent regions; temporal, spatial, and boundary priorities are instances of this principle. Using neural tangent kernel (NTK) dynamics, we theoretically analyze why standard PINNs do not obey this priority: their residual convergence order is governed by the NTK spectrum and is independent of the propagation path. Accordingly, we propose a unified multi-dimensional priority-constraint framework that partitions the domain along the propagation path and constructs negative-exponential residual weights, converting the physical propagation order into a training priority. For cases with coexisting priorities, we introduce a directional compatibility coefficient to clarify that "orthogonal directions can be coupled multiplicatively in synergy, whereas coaxial opposite directions cannot." Benchmark cases show that this method consistently improves the convergence behavior and prediction accuracy of PINNs on problems with clear propagation paths or constraint-dominated structures, without modifying the network architecture and with controllable additional computational cost.
