While mean field theory has provided remarkable insights into the behavior of neural networks in the infinite-width limit, you must recognize several phenomena in real-world deep learning that this theory does not adequately capture.

One major limitation is the inability of mean field theory to describe feature learning and the evolution of internal representations. In practical neural networks, especially those with finite width and depth, layers develop complex, hierarchical representations of data. These learned features are crucial for tasks such as image recognition and language understanding, but mean field theory, by focusing on distributional averages and assuming independence, largely overlooks these dynamic, emergent properties.

Another critical aspect not fully explained by mean field approaches is the intricate dynamics of representation learning during training. Real networks adapt their internal structure in response to data, often discovering abstract features that are not present in the input. Mean field theory, with its emphasis on statistical averages and limiting behavior, tends to treat activations and weights as random variables with fixed distributions, missing the nuanced, data-driven evolution of representations that occurs in practice.

There is also a significant gap between the predictions of infinite-width theory and the behavior of practical, finite-width neural networks. While the infinite-width limit allows for powerful mathematical simplifications—such as the emergence of Gaussian processes or the applicability of the neural tangent kernel—these results may not hold for networks of realistic size. Finite-width networks can exhibit behaviors such as:

• Feature reuse;
• Strong correlations between neurons;
• Nontrivial generalization patterns that are not predicted by mean field analysis.

Furthermore, practical networks are trained with stochastic optimization, data augmentation, and regularization techniques that introduce additional complexities beyond the scope of mean field models.

In summary, the boundaries of applicability for mean field results are defined by several key assumptions: infinite width, independence between neurons, and a focus on macroscopic averages rather than microscopic details. While these assumptions enable elegant theoretical results, they limit the ability of mean field theory to account for feature learning, representation dynamics, and many empirical phenomena observed in modern deep learning systems. Understanding these limitations is essential for interpreting the insights provided by mean field theory and for developing more refined models that bridge the gap between theory and practice.