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.