Dynamical coarse-graining of neural-network training

Learning in recurrent neural networks can fundamentally reshape their underlying dynamics, transforming initially chaotic activity into stable task-dependent behavior. We develop a non-equilibrium dynamical mean-field theory(DMFT) to describe this transition during learning. We show that a slow feedback-driven learning process generates an evolving effective feedback strength that drives the network through a transition from chaotic to stable dynamics defined by a bifurcation of the DMFT solution. By deriving the two-time correlation function throughout learning, we identify a critical feedback strength and a corresponding learning rate dependent critical time separating these regimes. The transition arises from the progressive deformation of an effective dynamical landscape by the growing learned feedback structure. Starting from the untrained state, the theory predicts the time evolution of the network output during training and shows quantitative agreement with numerical simulations.

We introduce Mode Dynamics, a framework that connects coordinated parameter motion during neural network learning to changes in model outputs. Starting from a recorded parameter trajectory, we decompose its updates into collective modes and map each mode to its state-dependent output effect. The framework organizes this link through three mechanisms. Decomposition identifies collective modes in parameter motion and their time-varying amplitudes. Reconstruction tests whether the effects induced by those modes recover local and accumulated output change. Concentration quantifies how behavioral contribution is distributed across modes. Controlled scalar regression and next-token experiments directly test these mechanisms. They show that small sets of leading modes reconstruct behavior, induced effects depend on the training path, and modal complexity varies with task and data. We then evaluate the framework on held-out image and language-model outputs using CIFAR-100 autoencoding and a public Pythia-70M pretraining trajectory. Across these settings, the results connect compact modal subspaces of recorded parameter motion to evolving behavior.

Understanding how stochastic gradient descent learns internal representations in wide neural networks trained on rich target functions is a central problem in machine learning. We study this question in two-layer neural networks learning target functions represented by networks of the same architecture, with arbitrary nonlinearity, and with network width and input dimension growing proportionally. In this regime, standard approaches for describing the learning dynamics require tracking a diverging number of overlaps, which measure correlations between learned and target features, leaving no obvious description in terms of a few order parameters. Here, we develop a mesoscopic description of learning by characterizing the distribution of overlaps. In the early stages of learning, we find that a Gaussian description of overlap statistics reduces the learning dynamics to a few collective variables that track learning of the target function's linear and quadratic components. We then estimate the timescale of specialization, when Gaussian closure breaks down and individual learned features begin to strongly correlate with specific target features. We find that specialization in extensive-rank networks is accelerated via a quadratic shielding mechanism, whereby learning the linear and quadratic components of the target function filters gradient fluctuations. We finally describe how the activation function determines the pattern of specialization and characterize the subsequent refinement of learned features. Together, these results show how collective learning gives way to individual feature discovery and clarify when learning in large neural networks admits a simple statistical description.