Identify NN-FTs that satisfy the Osterwalder–Schrader axioms

Determine which neural network field theory representations—specified by an architecture φ_θ and a parameter measure P(θ) that generate Euclidean Green’s functions—satisfy the Osterwalder–Schrader axioms (including reflection positivity) in dimensions d ≥ 2, thereby ensuring analytic continuation to unitary Lorentzian quantum field theories.

Background

The paper proves a universality theorem showing that any Euclidean quantum field theory, modeled as a probability distribution on tempered distributions S'(Rd), admits a neural network representation with countably many parameters (or formally with a single parameter). This provides existence but not physicality: Euclidean QFTs must satisfy the Osterwalder–Schrader (OS) axioms to guarantee analytic continuation to a unitary Lorentzian theory.

The authors note that in one dimension (quantum mechanics), mechanisms to engineer reflection positivity—one of the most subtle OS axioms—are known within neural network frameworks. However, they highlight that extending such constructions and verifying which higher-dimensional NN-FTs actually obey the OS axioms remains unresolved. Clarifying this would delineate the physically admissible subset of NN-FTs among the broad class guaranteed by the universality result.

References

Although we established that every GQS admits a NN description, including the distributions \mathcal{S}' ( \mathbb{R}d ) that provide the foundation of constructive field theory, it is not clear which NN-FTs obey appropriate physical conditions, such as the OS axioms.

— Universality of Neural Network Field Theory  (2601.14453 - Ferko et al., 20 Jan 2026) in Conclusion (Section 5)

We do not prove that our theory satisfies $OS3'$, for now we will assume that it does and will remind the reader whenever this assumption is used.

— Notes on algebraic perspective on neural network conformal theories  (2609.37074 - Leutheusser et al., 29 Sep 2026) in Section 4, subsection “Axiomatic Quantum Field Theories without Reflection Positivity”; Section 6, subsection “Lessons for neural-network conformal theories”

Whether an NN-FT construction can be reflection positive at finite width or at finite cutoff is open.

— Constructive Neural Network Field Theory: $φ_2^4$ in Finite Volume  (2610.00453 - Frank, 30 Sep 2026) in Section 3, Section 3.3 “The Osterwalder–Schrader axioms”