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On the physical interpretation of neural PDEs

Sauro Succi

Vol. 13 (2025), No. 3, 275–286
DOI: 10.2140/memocs.2025.13.275
Abstract

We highlight a formal and substantial analogy between machine learning (ML) algorithms and discrete dynamical systems (DDS) in relaxation form. The analogy offers a transparent interpretation of the weights in terms of physical information-propagation processes and identifies the model function of the forward ML step with the local attractor of the corresponding discrete dynamics. Besides improving the explainability of current ML applications, this analogy may also facilitate the development of a new class ML algorithms with a reduced number of weights.

Keywords
machine learning, partial differential equations, discrete dynamical systems
Mathematical Subject Classification
Primary: 35Axx, 44Axx, 68Txx
Milestones
Received: 9 February 2025
Accepted: 23 March 2025
Published: 6 September 2025

Communicated by Roberto Natalini
Authors
Sauro Succi
Italian Institute of Technology
00161 Rome
Italy
CNR-IAC
00185, Roma
Italy
Physics Department
Harvard University
Cambridge, MA
United States