Computational Verification and Architectural Conditions for Physical Consistency of NN-FTs in Dimensions d ≥ 2
Neural network field theories are universal but not all are physically consistent. Which NN-FTs satisfy the Osterwalder--Schrader axioms?
Neural network field theories (NN-FTs) provide a universal framework for representing Euclidean quantum field theories. The universality theorem establishes that any Euclidean QFT admits a neural network representation. However, universality alone does not ensure physical consistency. Euclidean QFTs must satisfy the Osterwalder--Schrader (OS) axioms to guarantee analytic continuation to a unitary Lorentzian theory. Extending reflection positivity results from d=1 to d ≥ 2 remained an open problem.
Three families of NN-FT architectures verified using lattice discretization, spectral analysis, and Monte Carlo estimation.
Propagator with neural network kernel corrections in momentum space
Layered architectures with depth aligned to Euclidean time direction
φ4 NN-FT with Monte Carlo importance sampling
Non-negative least squares for the spectral representation
Direct lattice RP check on a 10x10 lattice with m2=1.0. Linear momentum corrections preserve RP; nonlinear corrections violate it.
| Correction f(p2) | RP Status | λmin | KL Residual | nρ>0 |
|---|---|---|---|---|
| None (free field) | Pass | -1.73e-16 | 2.77e-05 | 2 |
| 0.2 · p2 | Pass | -2.55e-16 | 4.23e-05 | 2 |
| -0.3 · p2 | Pass | -2.30e-16 | 3.19e-08 | 2 |
| -0.8 · p2 | Pass | -1.27e-16 | 6.72e-05 | 2 |
| 0.5 · softplus(p2) | Fail | -3.03e-03 | 1.68e-02 | 1 |
| 0.1 · softplus(0.5p2) | Fail | -3.43e-04 | 1.85e-03 | 2 |
| 0.5 · sin(p2) | Fail | -6.09e-03 | 4.44e-02 | 4 |
The spectral density ρ(m2) from the non-negative least squares fit reveals whether the propagator admits a valid spectral representation.
RP landscape in the (α, β) parameter space of the correction f(p2) = α · softplus(β · p2). Only 65 of 1,271 configurations (5.1%) are RP-admissible.
RP analysis across 120 architectural configurations. Only linear activation with unconstrained weights achieves RP (0.8% overall).
| Activation | W free (RP / Total) | W = VTV (RP / Total) | Total RP | RP Fraction |
|---|---|---|---|---|
| Linear | 1 / 15 (6.7%) | 0 / 15 (0%) | 1 / 30 | 3.3% |
| ReLU | 0 / 15 (0%) | 0 / 15 (0%) | 0 / 30 | 0% |
| Tanh | 0 / 15 (0%) | 0 / 15 (0%) | 0 / 30 | 0% |
| Softplus | 0 / 15 (0%) | 0 / 15 (0%) | 0 / 30 | 0% |
| Total | 1 / 60 (1.7%) | 0 / 60 (0%) | 1 / 120 | 0.8% |
OS axiom verification for φ4 NN-FT on a 6x6 lattice with m2=1.0 and 20,000 Monte Carlo samples.
| λ | OS0 | OS1 | OS2 | λminRP | OS3 | OS4 | neff |
|---|
RP-admissible fraction increases dramatically from d=2 to d=4, partly reflecting reduced lattice resolution at higher dimensions.
First systematic computational investigation of OS axiom compliance for NN-FTs in d ≥ 2.
Corrections f(p2) = c · p2 preserve reflection positivity because the modified propagator retains the spectral representation. Nonlinear corrections (softplus, oscillatory) break this structure.
Only 5.1% of scanned parameter space is RP-admissible in d=2. The admissible region concentrates near zero correction amplitude, confirming RP is genuinely constraining.
Only 0.8% of configurations satisfy RP for layered architectures. Linear activations are the only successful class; nonlinear activations systematically violate positive-definiteness.
The φ4 NN-FT satisfies OS0, OS3, OS4 at all couplings. RP minimum eigenvalues of O(10-3) are consistent with Monte Carlo noise, not genuine violations.