Physics-informed neural networks (PINNs) often exhibit weight matrices that appear statistically random after training, yet their implications for signal propagation and stability remain unsatisfactorily understood, let alone the interpretability. In this work, we analyze the spectral and statistical properties of trained PINN weights using viscous and inviscid variants of the one-dimensional Burgers' equation, and show that the learned weights reside in a high-entropy regime consistent with predictions from random matrix theory. To investigate the dynamical consequences of such weight structures, we study the evolution of signal features inside a network through the lens of neural partial differential equations (neural PDEs). We show that random and structured weight matrices can be associated with specific discretizations of neural PDEs, and that the numerical stability of these discretizations governs the stability of signal propagation through the network. In particular, explicit unstable schemes lead to degraded signal evolution, whereas stable implicit and higher-order schemes yield well-behaved dynamics for the same underlying neural PDE. Our results offer an explicit example of how numerical stability and network architecture shape signal propagation in deep networks, in relation to random matrix and neural PDE descriptions in PINNs.

翻译：物理信息神经网络（Physics-informed neural networks, PINNs）训练后常呈现出统计意义上随机的权重矩阵，然而这些权重对信号传播与稳定性的影响尚未得到满意理解，更遑论其可解释性。本研究通过一维伯格斯方程的黏性及无黏变体，分析了训练后PINN权重的谱特性与统计性质，发现学习得到的权重处于与随机矩阵理论预测一致的高熵区域。为探究此类权重结构的动力学后果，我们借助神经偏微分方程（neural PDE）视角，研究网络内部信号特征的演化过程。研究表明，随机化权重矩阵与结构化权重矩阵可分别对应神经PDE的特定离散格式，且这些离散格式的数值稳定性决定了信号在网络中传播的稳定性。特别地，显式不稳定格式会导致信号演化退化，而稳定的隐式格式及高阶格式在相同神经PDE下能产生良好的动力学行为。本研究提供了数值稳定性与网络架构如何塑造深度网络中信号传播的具体实例，并建立了与PINN中随机矩阵及神经PDE描述之间的关联。