Pierfrancesco Urbani

CNRS Researcher in theoretical physics

My two main lines of research are: the physics of amorphous solids and high dimensional optimization problems, from learning to inference.

For what concerns structural glasses, my activity has focused on the construction of a theory of the glass phase starting from the soluble yet non-perturbative limit of infinite dimensions. This has allowed the characterization of the complete phase diagram of glasses. At low temperature, this analysis has shown the emergence of the so-called Gardner phase in which amorphous solids are marginally stable against external perturbations. This phase is the potential missing ingredient to unify the physics of anomalies of low temperature glasses and a large part of my recent activity has gone in this direction. The consequences of this discovery are already well established for colloidal-like glasses where we have shown that the Gardner phase is crucial for the computation of the critical exponent of the jamming transition and jamming-critical systems. My activity now focuses on the understanding of how marginal stability associated to the Gardner phase gives rise to non-linear excitations in amorphous solids and how they influence the rheology of these materials.

I also work in high dimensional optimization problems. My main focus has been the study of optimization algorithms and their properties in a set of applications, going from high-dimensional inference problems, to learning in simple neural networks. In high-dimensional statistical inference I have analyzed of the posterior measure of prototypical models first suggesting that reconstruction algorithms based on message passing could have been more effective than gradient based algorithms due to glassy phases. Then I focused on the study of gradient based algorithms first suggesting that the landscape trivialization transition is not a necessary condition for having good performances which has important consequences for learning problems where absence of spurious minima have been advocated to explain good performances. More recently my main activity has been the analysis of the main workhorse algorithm in deep learning which is stochastic gradient descent. I used dynamical mean field theory to analyze of this algorithm in simple yet prototypical cases and I have shown that SGD can outperform gradient descent in hard high-dimensional learning problems.

More recently I started a research line at the crossroad of statistical physics and theoretical neuroscience. This aims at understanding on the one hand collective learning phases of recurrent neural networks and, on the other, how bioologically plausible learning algorithms work.

Activity and results

Solution of structural glass models in infinite dimension:

Gardner transition in structural glasses: mean field theory, RG analysis and experiments.
Theory of the Jamming transition: computation of the critical exponents and characterization of marginal stability; jamming of non-spherical particles, upper critical dimension and hyperuniformity. Jamming critical phase in linear soft spheres. Avalanches. Rigidity transition in confluent tissues.
Theory of the low temperature harmonic excitations of amorphous solids. Euclidean Random Matrices, Boson Peak and localized soft excitations.
Theory of two level system excitations.
Microscopic theory of rheology of amorphous solids: yielding as a critical spinodal with emerging RFIM criticality, shear-jamming of stable hard sphere glasses, two-step yielding, stability map of glasses, breakdown of elasticity at low temperatures.
Dynamical Criticality: computation of the dynamical critical exponents at the MCT transition as well as at the Gardner point.
Quasi-equilibrium approach to the dynamics of structural glasses.

Constraint satisfaction and optimization problems

Jamming in multilayer neural networks.
Emergence of isostatic marginally stable critical phases both in infinite and finite dimensional optimization problems. Theory of the corresponding non-linear excitations.
Dynamical mean field theory for gradient-based optimization. Theory of Stochastic Gradient Descent.

High-dimensional statistical inference and machine learning

Glassy nature of the hard phase of inference problems.
Emergence of a generic gap between gradient based algorithms and message passing ones.
Analysis of gradient descent dynamics in high-dimensional inference problems. Momentum-based minimization algorithms.
Dynamical mean field theory for stochastic gradient descent. Persistent-SGD. Effective temperature.
Replica-symmetry-breaking implementation of Approximate Message Passing algorithms.

Learning in Recurrent Neural Networks

Tuning chaos through Hebbian forcing.
DMFT analysis of FORCE algorithm.
Generative modelling via chaotic dynamics

Disordered high dimensional optimal control

Field theory and renormalization

Field theory for the glass transition. Dynamical heterogeneities at the glass transition.
Non-perturbative renormalization group approach to instantons.
Field theory for soft anharmonic spin glasses in a field.
Scenario for the spin glass transition in a field: appearance of finite density of non-linear excitations.

Publications

G. Parisi, P. Urbani, F. Zamponi. Theory of simple glasses. Cambridge University Press 2020.

HDR manuscript, 2024, Slides

Google Scholars , Arxiv

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Lectures

IPhT Lectures on disordered and glassy systems (video)

Les Houches Lectures on dynamics in high dimension (2020) (lecture 1, lecture 2, lecture 3)

Lectures at the School on Disordered Elastic Systems, São Paulo, Brazil, 2022 (Lecture 1, Lecture 2, Lecture 3, Lecture 4)

IPhT Lectures: Generative models in a nutshell, upcoming next academic year, 2024/2025

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Conferences/Seminars/News