神经网络场论在计算机上可实现与不可实现的内容
What Neural Network Field Theory Can and Cannot Realise on a Computer
- Center for Theoretical Physics, Massachusetts Institute of Technology（麻省理工学院理论物理中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对神经网络场论提出适用于标准网络架构的不可能性定理，区分其四个版本，指出有限宽度系综的QFT/EFT解释存在缺陷，一维情形也难规避，仅可通过放弃有限方差或严格旋转不变性来突破。
AI中文摘要：
神经网络场论的一个目标是将量子场论（QFT）或有效场论（EFT）置于计算机上，以网络系综本身作为该理论。我们探究对于一类足够规则、可计算的函数，该目标能推进到何种程度。我们的主要结果是一个不可能性定理，其假设适用于标准网络架构。我们利用该定理，根据定义对象是有限宽度系综还是其无限宽度极限，以及目标是量子场论还是有效场论，对神经网络场论的四个版本进行区分。两种有限宽度解释均不直接一致：对于每点具有有限方差的有限宽度系综，量子场论解释不满足反射正性；而有效场论解释未建立尺度分离，使得正性违反无法被置于其有效性域之外。在两个极限版本中，一个可完全模拟，另一个仅能部分模拟，因为仅其平滑关联函数可通过可控误差计算。因此，在可控数值计算层面，量子场论和有效场论版本无法区分。一维情形可规避该阻碍，但对于余弦网络，在每一个有限宽度下反射正性仍被证明不成立。该定理仍有两种规避方式：放弃每点的有限方差或放弃严格旋转不变性，我们将讨论这两种可能性。
英文摘要：
One aim of neural network field theory is to put a quantum or effective field theory on a computer, with the network ensemble itself as the theory. We ask how far that aim can be pushed for a function class regular enough to be computed with. Our main result is a no-go theorem with assumptions that hold for standard network architectures. We use it to separate four versions of neural network field theory, according to whether the defining object is the finite width ensemble or its infinite width limit, and whether the target we want to compute is a quantum or an effective field theory. Neither finite width interpretation is straightforwardly consistent. For finite width ensembles with finite variance at each point, the QFT interpretation fails reflection positivity, while the EFT interpretation establishes no scale separation by which the positivity violation can be placed outside its domain of validity. Of the two limit versions, one can be simulated in full and the other only in part, as only its smeared correlators are computable with a controlled error. As such, at the level of a controlled numerical computation, the QFT and EFT versions cannot be distinguished. One dimension escapes the obstruction, yet reflection positivity is shown to still fail there at every finite width for the cosine network. Two escapes from the theorem remain, giving up either finite variance at a point or exact rotation invariance, and we discuss both of these possibilities.
