Tsallis statistics generalizes Boltzmann-Gibbs statistical mechanics through a single real parameter q that controls the weight assigned to rare and frequent events. Originally proposed to describe physical systems with long-range correlations, multifractal geometry, and heavy-tailed fluctuations, the framework has become a recurring ingredient in modern artificial intelligence (AI): it underlies sparse attention mechanisms (\textsc{sparsemax} and α-\textsc{entmax}), maximum-entropy reinforcement learning with controllable exploration, robust and heavy-tailed probabilistic models, and a family of generalized loss functions and regularizers. This paper offers a structured perspective on where Tsallis statistics meets AI. We first review the mathematical core: q-entropy and its variational (maximum-entropy) foundation, the q-exponential and q-logarithm, the q-central limit theorem, q-Gaussian distributions, and their dynamical origin in superstatistics, emphasizing the properties that matter for machine learning. We then survey applications across softmax generalization, reinforcement learning, sequential and graph neural models, generative and probabilistic modeling, loss design, and optimization, extracting the recurring design pattern in each case: a tunable interpolation between dense/uniform and sparse/peaked behavior governed by q. We further argue that the heavy-tailed weight spectra and gradient-noise statistics empirically observed in deep networks are themselves nonextensive signatures, placing modern learning dynamics within the scope of q-statistics. Finally, we discuss methodological pitfalls, the relationship to information geometry and q-exponential families, and open directions, arguing that q should be treated as a learnable inductive bias rather than a fixed hyperparameter.
Deep learning has managed to evade numerous intuitions from classical statistics to achieve unprecedented performance on a number of real-world tasks. In this article, we investigate the key features and surprises of deep learning from a physics-informed perspective, taking care to point out and justify where possible the many choices inherent in constructing a deep learning model. In particular, we review the phenomenon of neural scaling laws and discuss their interplay with the constraints and inductive biases which may be present when applying machine learning to problems in physics.
Normally the statistical mechanics of learning treats constraints on weight distributions as restrictions that shrink the space of possible solutions. Therefore, it reduces model capacity. In this paper we would like to take a contrary approach, which, however, is based on the earlier work on distribution-constrained perceptrons. Rather than treating a prescribed weight distribution as a mere restriction, we propose that it defines the intrinsic geometry upon which learning naturally unfolds. We formulate both deep neural networks and variational quantum circuits as gradient flows on a product of Wasserstein manifolds -- one classical Wasserstein space for each layer and one quantum Wasserstein space for the circuit parameters. Within this geometry, the capacity reduction, which was previously associated with distributional constraints, appears as the metric structure of the constraint manifold itself. We develop a hierarchical mean-field description for deep networks, extend the framework to the quantum setting using the quantum Wasserstein distance of order 1, and introduce two such practical algorithms, Hierarchical DisCo-SGD and Quantum DisCo, that follow approximate geodesics on the manifold of the product itself. Experiments on teacher-student problems, standard image classification tasks, and small variational quantum classifiers show that respecting these distributional geometries improves generalization, stabilizes training, and reduces the severity of barren plateaus compared with unconstrained and purely norm-based baselines. This approach firstly reframes structural constraints as geometric priors and suggests a route for incorporating biological, spectral, or hardware-derived distributional information into both learning systems, viz., classical and quantum learning.
Deep learning is increasingly viewed as a dynamical process in parameter space, yet many existing theories still treat training as a closed optimization system. This view is limited for real-world AI, where models operate under uncertainty, resource constraints, distribution shift, downstream decision risks, and human feedback. We propose Human-Centered Learning Mechanics (HCLM), a dynamical and information-theoretic framework for open and controlled learning systems. The central idea is that entropy regularization is useful only when the chosen entropy surrogate generates a non-degenerate information force along the optimization trajectory. Otherwise, entropy terms may produce weak, unstable, or misaligned gradients, causing the dynamics to collapse toward ordinary loss minimization. We introduce the notion of effective entropy and study tractable geometric entropy surrogates, including variance-based and log-determinant covariance proxies. The paper makes three contributions. First, it formalizes entropy regularization through effective information force and characterizes degenerate entropy regimes. Second, it derives convergence, entropy-flow, Wasserstein-gradient-flow, and noisy-representation generalization results under explicit assumptions. Third, it offers a conditional dynamical interpretation of scaling-law-like behavior as a balance between information injection, entropy dissipation, and residual risk, without claiming an unconditional derivation of empirical neural scaling laws. Controlled representation-learning experiments support the hypothesis that geometric entropy surrogates, especially log-determinant covariance entropy, induce stronger and more stable information forces than softmax-normalized entropy.