https://doi.org/10.1140/epjp/s13360-026-07549-0
Regular Article
Randomness and signal propagation in physics-informed neural networks (PINNs): a neural PDE perspective
1
Department of Civil, Computer Science and Aeronautical Technologies Engineering, Università degli Studi Roma Tre, via Vito Volterra 62, 00146, Rome, RM, Italy
2
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, 560064, Bangalore, Karnataka, India
3
Center for Life Nano-& Neuro-Science, Italian Institute of Technology (IIT), viale Regina Elena 295, 00161, Rome, RM, Italy
a
This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
16
September
2025
Accepted:
6
March
2026
Published online:
22
March
2026
Abstract
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.
Copyright comment Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
