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Neural Network Field Theory at Finite Width
Christian Ferko, Aaron Mutchler
TL;DR
The paper asks what properties of exact countably parameterized neural-network representations survive when replaced by finite-width networks. It examines finite-parameter NN-QM and NN-FT models and finds that finite width typically sacrifices at least one conventional Euclidean QFT property.
Problem
The paper asks which features of countably infinite neural-network representations remain exact or approximate, and which must be relaxed, at finite width.
Method
The paper investigates finite-parameter neural-network representations of quantum mechanical models and quantum field theories through complementary analyses of their preserved and violated properties.
Results
For N < ∞, neural-network representations typically violate at least one conventional Euclidean QFT property, such as reflection positivity or cluster decomposition.
Takeaways & Limitations
Finite-width truncation constrains which conventional Euclidean QFT properties can be preserved, even though neural-network representations can reproduce all correlation functions in the stated construction.
Takeaways & Limitations
Cases involving an indefinite weight fall outside the scope of the present investigation.
Abstract
from arXiv · showhide
Under mild assumptions, any quantum mechanical (QM) model or quantum field theory (QFT) admits a representation in terms of an ensemble of neural networks with countably many random parameters. We investigate the features of NN-QM and NN-FT models with finitely many parameters, such as a feedforward network of width $N < \infty$. We find that, generically, such models must violate one of the properties of conventional Euclidean QFTs, such as reflection positivity or cluster decomposition. We present several complementary ways of understanding which features can and cannot be preserved at finite $N$, both in QM and in QFT.
1 Introduction
The paper asks which properties of quantum theories survive when countably infinite neural-network realizations are replaced by finite-parameter architectures. It characterizes architecture- and assumption-dependent obstructions in quantum mechanics and quantum field theory, finding generic failures of conventional Euclidean-QFT properties at finite width.
- Research question: Finite-parameter networks are studied as replacements for countably infinite neural-network realizations of quantum mechanics and quantum field theories.Under mild assumptions, one-dimensional and d-dimensional theories with suitable measures admit countably parameterized neural-network realizations.
- Research question: The central question is which properties of the underlying quantum theory can be realized exactly, survive approximately, or must be relaxed at finite N.The paper uses this finite-N question as its organizing perspective.
- Scope of finiteness: A finite parameter count alone does not ensure a genuinely finite network, because a single parameter can encode an arbitrary probability law through a measurable pushforward.Such encodings may contain unbounded information in infinitely many digits and require infinitely many operations to evaluate.
- Architectures: The analysis distinguishes architecture-agnostic statements from results for natural finite-width architectures, including single-layer networks with finitely many parameters.The representative architecture uses parameters θ ∈ {w_i, a_i, b_i}, allows varied neurons, and is restricted to one layer for simplicity.
- Quantum mechanics and QFT: In quantum mechanics, finite sums of piecewise-differentiable activations cannot generate the nowhere-differentiable paths supporting standard path-integral measures.In higher-dimensional QFT, typical fields are proper Schwartz distributions rather than pointwise functions, motivating separate finite-width analyses.
- Main findings: Generically, finite-width models fail to satisfy all Osterwalder-Schrader axioms, including reflection positivity and cluster decomposition.The paper also identifies finite-N networks that cannot reproduce coincident-point divergences, while discussing examples that can produce them.
- Main findings: Finite-N realizations can introduce non-local interactions scaling as 1/N, providing another reason they cannot reproduce all correlation functions of conventional local quantum theories.The paper presents complementary results on divergences, OS axioms, and broader neuron classes.
2 Neural Network Quantum Mechanics at Finite N
NN-QM represents a stochastic process through random parameters and deterministic functions, with correlation functions defining the realization. At finite width, non-trivial quantum-mechanical processes cannot be reproduced exactly, while KKL truncation gives the optimal finite-dimensional approximation.
- An NN-QM pushes a parameter law forward to an ensemble of trajectories whose correlation functions match those of a Euclidean quantum system.
- The KKL construction represents the process with countably many random coefficients multiplying deterministic eigenfunctions.
- Finite-width architectures cannot reproduce the connected two-point function of a non-trivial Osterwalder-Schrader process, and therefore cannot realize it exactly.
- Dimension-Counting: The obstruction reflects a dimensional mismatch: a finite network cannot fluctuate throughout the whole L2(T) space required by the quantum representation.
- KKL Optimality: The first N KKL modes maximize captured variance among N-dimensional feature spaces and minimize integrated mean-square truncation error.
- KKL Optimality: EN ≥ λN+1 provides a basis-independent lower bound for every finite-N approximation, while the full omitted-eigenvalue sum gives the sharp KKL bound.
3 Neural Network Quantum Field Theory at Finite N
Finite-width neural-network QFT realizations can match selected two-point behavior, but finite-rank, variance, and momentum-support obstructions prevent generic agreement with non-trivial QFTs. Preserving some properties therefore forces violations of others, including Osterwalder–Schrader axioms or cluster decomposition.
- Osterwalder–Schrader constraints: For d ≥2, coincident-point divergences in a finite-width realization require violating at least one of temperedness, Euclidean invariance, or reflection positivity.The conclusion is stated for theories with a non-trivial Källén–Lehmann spectral measure.
- Coincident-point limits: Finite networks have bounded mollified coincident variance, so they cannot reproduce the divergent behavior required by non-trivial Källén–Lehmann representations.This obstruction applies broadly to finite networks with finite moments.
- Infinite-variance examples: Sufficiently singular parameter statistics can produce coincident-point divergences and match a reflection-positive two-point function at finite width.The single-neuron construction preserves the divergent variances of point fields while matching smeared two-point functions.
- Infinite-variance examples: Finite-width networks can match a target two-point function while generally failing reflection positivity for higher-point functions.The paper gives the example of reproducing ⟨x(t)x(s)⟩ = ce^−m|t−s| but not its higher-point functions.
- Momentum-space analysis: Cluster decomposition conflicts with finite-network momentum support because typical clustered QFT draws have full Fourier support, whereas finite-network draws occupy finitely many momentum lines.The resulting field-configurational measures cannot agree, yielding a momentum-space obstruction independent of the finite-variance argument.
- Distributional neurons: Distributional neurons remove the function-versus-distribution mismatch but retain the finite-rank obstruction, leaving infinitely many deterministic directions.A finite truncation therefore cannot reproduce the correlation functions of a non-trivial QFT with a Källén–Lehmann representation.
4 Conclusion
At finite width, neural-network representations generically sacrifice at least one hallmark of conventional quantum theories, while the paper identifies several directions for characterizing and extending these results.
- Finite-N neural network representations typically violate at least one Osterwalder–Schrader property, such as reflection positivity or cluster decomposition.
- Exactly realizing a conventional QM or QFT representation requires an infinite-width limit or an architecture and parameter density that evade the theorem assumptions.
- Finite-N models may nevertheless be interesting because they can be more exotic than familiar quantum models.
- Future work could replace qualitative obstruction theorems with quantitative bounds describing axiom violations as functions of width N.
- A complexity-theoretic definition could distinguish genuinely finite architectures from encodings that hide infinitely many operations in real-number digits.
- The analysis is presently limited to theories defined by real non-negative measures, excluding sign-problem models such as theories with fermions or gauge fields.