# Randomness and signal propagation in physics-informed neural networks (PINNs): A neural PDE perspective

arXiv: https://arxiv.org/abs/2509.18131 · HTML: https://mlsanity.com/paper/2509.18131/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 20 references that are in the map.
- First posted on arXiv 12 Sep 2025.

## Builds on (most cited first, 8 of 8)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning](https://mlsanity.com/paper/1810.01075/) (2018)
- [A Random Matrix Approach to Neural Networks](https://mlsanity.com/paper/1702.05419/) (2017)
- [Random matrix analysis of deep neural network weight matrices](https://mlsanity.com/paper/2203.14661/) (2022)
- [Impact of classification difficulty on the weight matrices spectra in Deep Learning and application to early-stopping](https://mlsanity.com/paper/2111.13331/) (2021)
- [Connections between Numerical Algorithms for PDEs and Neural Networks](https://mlsanity.com/paper/2107.14742/) (2021)
- [A note on the physical interpretation of neural PDE's](https://mlsanity.com/paper/2502.06739/) (2025)

## Built on by (most cited first, 1 of 1)

- [Deep Neural Networks as Discrete Dynamical Systems: Implications for Physics-Informed Learning](https://mlsanity.com/paper/2601.00473/) (2026)

## Abstract (arXiv)

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.