Research Topics

#### Project

Grant-in-Aid for Scientific Research, MEXT, Japan "Physics of Glass and Jamming Transitions Captured by Spins: from Soft Matter Physics to Statistical Physics of Information Processing" (No. 19H01812) 2019-2021

Grant-in Aid for Scientific Research 24540403, MEXT, Japan 2012-2015

Grant-in Aid for Scientific Research, MEXT, Japan 2015-2017

#### Glass Physics : From Condensed Matters to Deep Neural Networks

In crystals, the solid states with periodic arrangements of molecules exhibit rigidity. Glasses are as disordered as liquids but rigid. The cause for the emergence of the rigidity in glasses has been recognized as a long standing, one of the most important open problems in physics. Recently mechanical properties of jammed states, i. e. densly packed states of particles with macroscopic sizes, such as colloids, emulsions and sands attracts much attention of physicists. We are studying these problems using theoretical methods such as the replicated liquid theory and numerical simulations such as molecular dynamic (MD) simulations.

Replicated liquid theory combines the density functional theory of liquids and the replica method, which has been developed in the statistical mechanical studies of disorderd systems. Quite recently it has been shown that mean-field theories which become exact in the large dimensional limit $d \to \infty$ (Charbonneau-Kurchan-Parisi-Urbani-Zamponi 2014) can be constructed based on this approach. Moreover it has been shown that on top of the usual 1 step replica symmetry breaking (RSB) which accompanies the glass transition, continous RSB (Gardner's transition) takes place at higher densities approaching the jamming density and dominates the critical properties of the system around the jamming.

Our recents results: Quasi-static response of the glasses under shear around the glass and jamming transitions

• We showed that, somewhat paradoxically, the rigity, i.e. the shear-modulus of a piece of a glass can be computed twisting it in the liquid state (!) using the replica method (Yoshino-Mezard 2010)A (Yoshino 2012)
• We analyzed the anomalous scaling properties of the shear-modulus at the glass and jamming transitions of hardsphere glasses exactly in the large-dimensional limit. We showed that the rigity itself becomes hierarhical refecting the continous RSB by the Gardner's transition at higher densities. (Yoshino-Zamponi 2014)
• Using the "state following method" in the replicated liquid theory, we analyzed the evolution of metastable glassy states under shear (and compression) from the linear response regime up to its failure, i.e. yielding. (Rainone-Urbani-Yoshino-Zamponi 2015)

We are currently investigating rheological properties of these systems performing MD simulations looking for aniticipated non-trivial effects implied by the theoretical results. Large scale MD simulations are performed using supercomputing facilities of our institute (Cybermedia Center,Osaka Univ. ), ISSP Univ. of Tokyo and Inst. for Molecular Science (IMS).

• Protocol-dependent shear modulus of amorphous solids (Nakayama-Yoshino-Zamponi 2016)
• Exploring the complex free energy landscape of the simplest glass by rheology (Jin-Yoshino 2017)
• A stability-reversibility map unifies elasticity, plasticity, yielding, and jamming in hard sphere glasses (Jin-Urbani-Zamponi-Yoshino 2018)
• #### Beyond simple hard spheres - glass transitions and jamming of rotational degrees of freedom

Patchy colloids, elipsoidal colloids and most of real molecules have rotational degrees of freedom in addition to the translational degrees of freedom. We are currently investigating theoretically systems of such non-spherical particles by constructing exactly solvable models in a large dimensional limit (Yoshino 2018, preprint)

#### Disorder-free spinglasses

Do spin systems in "supercooled paramagnetic states" exibit glass transitions, without quenched disorder much like the structural glasses but unlike the usual spinglasses with quenched disorder? We showed the answer is "YES" by constucting and analyzing explicitely a family of exactly solvable vectorial spin models with large number $M \to \infty$ of spin components (Yoshino 2018, SciPost Physics)

Recently we are working on a frustrated magnet on a pyroclore lattice and found signatures of spin glass transitions without quenched disorder. (Mitsumoto-Hotta-Yoshino 2019, PRL)

#### Vectorial constraint satisfaction problems and statistical inference problems

Moreover (Yoshino 2018, SciPost Physics) is related to constrained satisfaction problems of vectorial degrees of freedom (such as continuous coloring) and related statistical inference problems.

With M=2 component spins (XY spins) interacting with each other via p=2 body hard-core potential in the spin space, we can model the continuous coloring problem. Here dynamical variables are "color angle" in the range $0 < \theta < 2\pi$ (HSVY color map) put on nodes. The color angles on adjacent nodes must be separated in the angle larger than a threshold value. Increasing the connectivity and/or increasing the threshold value, the strength of the constraint increases. This leads to clustering of the solution space (glass transition), emergence of hierarchy in the solution space (continuous replica symmetry breaking) and eventually vanishement of the solution space (SAT/UNSTAT transition, jamming).

#### Why Deep Neural Networks actually work?

Natural extension of the above work is to analyze deep neural networks (DNN). In DNNs, there are not much 'quenched disorder' except for those at the input/output boundaries. Then the approach of (Yoshino 2018, SciPost Physics) becomes useful. We analyzed the solution space of DNNs constructing a replica theory and performing numerical simulations (Yoshino 2020, SciPost Physics Core)

. Our results provide proposals to solve the following issues.

H. Yoshino, "From complex glass to simple liquid: layering transitions in deep neural networks", 40 years of Replica Symmetry Breaking (2019/09@Sapienza Univ. of Rome) talk(slide+video)

Deep Neural Network and Physics 2019@Kyoto (DLAP2019) DLAP2019

Machine learning by deep neural networks (DNN) is successful in numerous applications. However it remains challenging to understand why DNNs actually work so well. Given the enormous parameter space, which is typically orders of magnitude larger than that of the data space, and the flexibility of non-linear functions used in DNNs, it is NOT very surprising that they can express complex data. What IS surprising is that such extreme machines can be put under control. On one hand, one would naturally fear that learning such huge number of parameters would be extremely time consuming because the fitness landscape is presumably quite complex with many local traps. Moreover over-fitting or poor generalization ability seem unavoidable in such over-parametrized machines. One would not dare to fit a data set of 10 points by a 100 th order polynomial(!), which does not make sense usually. Quite unexpectedly, these issues seem to be somehow resolved in practice and such extreme machines turned out to be very useful. Thus it is a very interesting scientific problem to uncover what is going on in DNNs. This is also important in practice because we wish to use DNNs not merely as mysterious black boxes but control/design them in rational ways.

To tackle this problem, we develop a statistical mechanical approach based on the replica method to study the solution space of deep neural networks. Specifically we analyze the configuration space of the synaptic weights in a simple feed-forward perceptron network within a Gaussian approximation for two scenarios : a setting with random inputs/outputs and a teacher-student setting. By increasing the strength of constraints,~i.~e. increasing the number of imposed patterns, successive 2nd order glass transition (random inputs/outputs) or 2nd order crystalline transition (teacher-student setting) take place place layer-by-layer starting next to the inputs/outputs boundaries going deeper into the bulk. For deep enough network the central part of the network remains in the liquid phase. We argue that in systems of finite width, weak bias field remain in the central part and plays the role of a symmetry breaking field which connects the opposite sides of the system. In the setting with random inputs/outputs, the successive glass transitions bring about a hierarchical free-energy landscape which evolves in space: it is most complex close to the boundaries but becomes renormalized into progressively simpler one in deeper layers.

#### Nonlinear rheology, nonlinear electric transports..

We studied some interesting problems of Josephson junction arrays (JJA) under magnetic field, which is related to the problem of frustrated magnets, jamming transition, frictional transition, non-linear rheology.... We analyzed the problems using theoretical and numerical approaches.

imovie) JJA under magnetic field driven by external electric current: the pattern of vorticies, electric currents (vertical/horizontal). from (Yoshino-Nogawa-Kim 2009)

#### Spinglasses and related systems

Spiglasses and a class of related systems exibit glassy phases with marginal stability: the stable confiurations of the systems may change drastically under tiny variations of parameters such as the temperature. Combined with the slow dynamics, non-trivial dynamical phenomena such as the rejuvenation-memory effects can take place. We analyzed the problem by theoretical, numerical approaches and also in some collobration with experimental groups.

A phenomenological theory of the memory effect (Yoshino-Lemaitre-Bouchaud 2001)

A collobration with an experimental group (Jonsson-Mathieu-Nordblad-Yoshino-Katori-Ito 2004)

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