Abstract
The project plans to go beyond the paradigm of independent identically distributed (i.i.d.) disorder that the theory of complex systems has followed so far. From its birth in the mid seventies, to the Parisi theory of the spin glass, up to the recent rigorous mathematical results, such theory has been mostly developed with the hypothesis of i.i.d. random variables appearing in the Hamiltonian function. While such a framework represents a natural and physically motivated starting point to investigate magnetic alloys, nowadays the most striking applications of disordered systems, such as those coming from machine learning and inference, require to go beyond. Machine learning indeed, or more generally a high dimensional statistical inference task, can be interpreted as the solution of a statistical mechanics system of interacting particles with random parameters, usually called Boltzmann Machine. The data, represented by a suitable distribution, is in fact mapped into a distribution of those parameters that in general have no reason to be neither independent nor identically distributed. More precisely our plan is to proceed along three different and somehow complementary themes. Multispecies structure Beyond the identical distribution assumption: we want to avoid the use of the full permutation symmetry among particles and we propose to break it into different groups. This leads to the multispecies structure where, in particular, we will focus on the non convex setting that contains the case of deep architectures. Multiscale measure Beyond the independent distribution assumption: we plan to investigate a correlated noise structure called multiscale measure, introduced within the rigorous approaches to the Parisi solution. We aim to analyse multispecies models endowed with a multiscale structure from the static and dynamic point of view. In particular we are interested in out of equilibrium properties like aging and violation of the Fluctuation Dissipation in relation to machine learning dynamics. Orthogonally invariant noise Again beyond the independent distribution assumption: we address spin-glass models with orthogonally invariant noise drawn from Orthogonal Random Matrix Ensembles, with a particular emphasis on the inference task of retrieving a finite rank matrix corrupted by orthogonally invariant noise with methods borrowed from the Statistical Mechanics of Disordered Systems. The above generalizations of the standard Boltzmann Machines and the investigation of their behavior in relation with data structure, network architecture and weights regularization, will provide a closer look at learning and inference regimes through the detection of phase transitions. We believe that the proposed research project represents a fundamental step toward the identification of the still lacking theoretical foundation of artificial intelligence. That in turn will favor, in perspective, progress with awareness and responsibility toward its use in society. Achieved Results
Results achieved
On multispecies and non-convex spin-glass models, the project advanced the theory of heterogeneous mean-field systems, in which variables aregrouped into interacting populations representing modular architectures such as layered neural networks. New results on limit theorems, phase diagrams and fluctuation properties were obtained on complete graphs (Contucci, Mingione, Osabutey, Ann. Henri Poincaré 2024; Camilli, Mingione, Osabutey, Mathematics 2025) and diluted random graphs (Alberici, Contucci, Mingione, Zimmaro, arXiv 2024), showing that multispecies structures generate richer phase diagrams than homogeneous systems. The thermodynamic structure of these models was also used to optimize deep learning architectures (Branchini, Contucci, Cormenier, Manzan, npj Artificial Intelligence 2025), and a companion study showed how multispecies structures affect consensus and polarization in opinion dynamics on modular networks (Zimmaro, Contucci, Kertész, Entropy 2023). On multiscale disordered systems, inspired by the hierarchical organization of spin-glass theory, the project established that multiscale measures naturally induce a gap in the distribution of the order parameter, with direct implications for training dynamics governed by multiple effective time scales (Camilli, Contucci, Mingione, Tantari, Ann. Henri Poincaré 2025). Convergence conditions were derived for stochastic dynamics with widely separated time scales and temperatures (Alberici, Macris, Mingione, Ann. Henri Poincaré 2024), and a rigorous probabilistic origin for multiscale measures was established, rooted in entropic constraints and reinforced stochastic processes (Camilli, Contucci, Mingione, arXiv 2025). On high-dimensional inference with structured noise, the project proved a central limit theorem for the overlaps of the Wigner spiked model on the Nishimori line (Camilli, Contucci, Mingione, Electron. J. Probab. 2025) and obtained a variational representation of the mutual information for a mismatched inference problem — where the statistician has only partial knowledge of the signal-generating mechanism — revealing a glassy, Parisi-type complexity phase in high-dimensional inference (Camilli, Contucci, Mingione, Entropy 2024). These tools were applied to an ECG-based machine-learning model for mortality risk assessment in a large European cohort (Doneda, Lanzarone, Giberti, Vernia, Vjerdha, Giovanardi, J. Electrocardiology 2025), showing the relevance of non-i.i.d. inference theory to real-world biomedical data. Beyond the original scope, the theoretical tools developed were extended to structured symbolic data, through the study of the arithmetic structure of natural numbers via rooted-tree representations: new asymptotic density formulas (Conti, Contucci, Iudelevich, Ann. Univ. Ferrara 2024; arXiv 2025), long-range statistical correlations in the rooted-tree encoding of integers (Contucci, Giberti, Osabutey, Vernia, Physica A 2026), and a test of whether transformer architectures can learn this deterministic sequence (Breccia, Gerace, Lippi, Sicuro, Contucci, arXiv 2025). Further interdisciplinary applications included statistical-mechanics models of peer-to-peer effects in pharmaceutical R&D; investment networks (Ferrari, Fedele, Luzi, Vernia, Physica A 2025) and synchronization phenomena via degenerate Kolmogorov–Fokker–Planck equations (Pecorella, Polidoro, Vernia, J. Math. Anal. Appl. 2025). The project also motivated a reflection on the energetic sustainability of artificial intelligence through the "Economic Productivity of Energy" indicator (Contucci, Osabutey, Zimmaro, Energies 2026), complemented by an encyclopedic entry on artificial intelligence (Contucci, Enciclopedia Italiana Treccani, 2024). Overall, the results significantly advanced the mathematical theory of multispecies systems, multiscale disorder, and high-dimensional inference beyond the i.i.d. paradigm, opening new interdisciplinary directions connecting statistical mechanics with artificial intelligence, symbolic structures, biomedical risk modelling, and sustainability. The results have been published in leading international peer-reviewed journals, including Annales Henri Poincaré, Electronic Journal of Probability, Entropy, Mathematics, Physica A, and Energies, and circulated as preprints on arXiv, ensuring broad and open access. The outcomes were also presented at international seminars and conferences, including two events organized by the team: "ROCKIN' AI 2024" (Roccella Ionica, September 2024) and "Mathematical Physics and Beyond" (Bologna, May 2024), promoting dialogue between statistical physics, mathematics and machine learning and the participation of young researchers. The project also engaged in public outreach, including an invited talk at the Festival "Scienza e Filosofia" in Foligno and educational video lectures for RAI Scuola, contributing to a scientifically grounded public understanding of artificial intelligence.Dettagli del progetto
Responsabile scientifico: Pierluigi Contucci
Strutture Unibo coinvolte:
Dipartimento di Matematica
Coordinatore:
ALMA MATER STUDIORUM - Università di Bologna(Italy)
Contributo totale di progetto: Euro (EUR) 170.630,00
Contributo totale Unibo: Euro (EUR) 96.600,00
Durata del progetto in mesi: 24
Data di inizio
28/09/2023
Data di fine:
28/02/2026
