Universality criteria for broken-isomorphism physical neural networks

Identify the necessary and sufficient architectural and dynamical features that enable universal computation or universal function approximation in broken-isomorphism physical neural networks, i.e., analog systems trained directly in their native physics without enforcing operation-by-operation mathematical isomorphism to digital neural networks.

Background

Broken-isomorphism physical neural networks (PNNs) depart from strict, operation-by-operation emulation of digital neural networks and instead train the native physical transformations of the underlying hardware. This paradigm has the potential to deliver major gains in speed and energy efficiency by leveraging the natural dynamics of physical systems.

A central theoretical gap is understanding what properties of such physical systems are required to approximate arbitrary functions or to perform universal computation. Clarifying these requirements would guide the design of scalable, high-performance PNN hardware and training strategies.

References

One complication with broken-isomorphism PNNs is that it is often unknown what features are required for universal computation or universal function approximation.

— Training of Physical Neural Networks  (2406.03372 - Momeni et al., 2024) in Box1: PNNs

The flexibility of the paradigm leaves open further constructions: with a continuous value set W, for instance, the same scheme may provide a substrate for analog computing. We currently offer no proof or formal guarantee in that direction.

— Emergent Models: Intelligence from Tiny Substrates  (2608.14019 - Bocchese et al., 14 Aug 2026) in Section 2.4, immediately following Theorem 1

It does, however, reframe a question we cannot yet answer: whether the better route to intelligence on silicon is to build highly simplified models of biological neural networks, as artificial neural networks do, or to construct an artificial physics from which arbitrary structure can emerge.

— Emergent Models: Intelligence from Tiny Substrates  (2608.14019 - Bocchese et al., 14 Aug 2026) in Conclusion, paragraph beginning “A second motivation”

More broadly, an open question is whether excitonic networks are universal approximators in the limit $N\rightarrow\infty$, given that the degree of the rational input-output relation grows without bound with system size and rational functions are known to approximate a broad class of functions.

— Coherent advantage in the computational expressivity of excitonic networks  (2609.10448 - Du et al., 9 Sep 2026) in Section Conclusions