New research proves DNN training equals renormalization group method

A new theoretical paper by Fuzhou Gong and Zigeng Xia (arXiv:2606.00157) establishes a formal correspondence between the training of fully connected deep neural networks (FCDNNs) and the renormalization group (RG) method from statistical physics. The authors previously proved this equivalence using discrete input data (1D Ising model). Now they generalize to continuous data drawn from the exponential family (including Gaussian, Poisson, and Bernoulli distributions). Their key result: after optimal training, the characteristic parameters of the feature layer output equal the fixed points of the input data's characteristic parameters under the RG transformation for continuous fields. This means the network's hidden layers perform an RG-like coarse-graining, automatically discovering the most relevant features while discarding noise.

The paper provides a concrete mathematical framework that explains DNNs' remarkable ability to extract main features — a process that previously lacked rigorous theoretical underpinnings. By showing that training and RG are equivalent operations, the authors offer a unified explanation for deep learning's success on real-world data. The work also opens the door to using physics-inspired methods to analyze, debug, and potentially design more interpretable neural architectures. While experimental validation on complex datasets (e.g., images, text) remains future work, this theoretical advance has already drawn attention from the ML and physics communities for bridging two previously separate domains.