有限宽度神经网络无法完全保持量子场论的对称性与关联性质。
Neural Network Field Theory at Finite Width
- 用有限参数神经网络表示量子力学与场论模型
- 发现有限宽度下必然违反反射正定性等核心性质
- 揭示了深度学习模型与物理理论的本质差异
在温和假设下，任意量子力学（QM）模型或量子场论（QFT）均可表示为一组具有可数随机参数的神经网络。本文研究了参数数量有限（如宽度 $N < ∞$ 的前馈网络）时的神经网络-量子模型（NN-QM）与神经网络-场论（NN-FT）特性。结果表明，这类模型通常必须违背经典欧几里得量子场论的核心性质，如反射正定性或簇分解。我们通过多种互补方法分析了在有限 $N$ 下哪些特征可保留、哪些不可保留，涵盖量子力学与量子场论情形。
原文摘要 · Abstract (English)
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.
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