Machine learning without a processor: Emergent learning in a nonlinear analog network.

Standard deep learning algorithms require differentiating large nonlinear networks, a process that is slow and power-hungry. Electronic contrastive local learning networks (CLLNs) offer potentially fast, efficient, and fault-tolerant hardware for analog machine learning, but existing implementations are linear, severely limiting their capabilities. These systems differ significantly from artificial neural networks as well as the brain, so the feasibility and utility of incorporating nonlinear elements have not been explored. Here, we introduce a nonlinear CLLN-an analog electronic network made of self-adjusting nonlinear resistive elements based on transistors. We demonstrate that the system learns tasks unachievable in linear systems, including XOR (exclusive or) and nonlinear regression, without a computer. We find our decentralized system reduces modes of training error in order (mean, slope, curvature), similar to spectral bias in artificial neural networks. The circuitry is robust to damage, retrainable in seconds, and performs learned tasks in microseconds while dissipating only picojoules of energy across each transistor. This suggests enormous potential for fast, low-power computing in edge systems like sensors, robotic controllers, and medical devices, as well as manufacturability at scale for performing and studying emergent learning.

Discontinuous instabilities in disordered solids.

Under a sufficiently large load, a solid material will flow via rearrangements, where particles change neighbors. Such plasticity is most easily described in the athermal, quasistatic limit of zero temperature and infinitesimal loading rate, where rearrangements occur only when the system becomes mechanically unstable. For disordered solids, the instabilities marking the onset of rearrangements have long been believed to be fold instabilities, in which an energy barrier disappears and the frequency of a normal mode of vibration vanishes continuously. Here, we report that there exists another, anomalous, type of instability caused by the breaking of a "stabilizing bond," whose removal creates an unstable vibrational mode. For commonly studied systems, such as those with harmonic finite-range interparticle interactions, such "discontinuous instabilities" are not only inevitable, they often dominate the modes of failure. Stabilizing bonds are a subset of all the bonds in the system and are prevalent in disordered solids generally. Although they do not trigger discontinuous instabilities in systems with vanishing stiffness at the interaction cutoff, they are, even in those cases, local indicators of incipient mechanical failure. They therefore provide an accurate structural predictor of instabilities not only of the discontinuous type but of the fold type as well.

Identifying microscopic factors that influence ductility in disordered solids.

There are empirical strategies for tuning the degree of strain localization in disordered solids, but they are system-specific and no theoretical framework explains their effectiveness or limitations. Here, we study three model disordered solids: a simulated atomic glass, an experimental granular packing, and a simulated polymer glass. We tune each system using a different strategy to exhibit two different degrees of strain localization. In tandem, we construct structuro-elastoplastic (StEP) models, which reduce descriptions of the systems to a few microscopic features that control strain localization, using a machine learning-based descriptor, softness, to represent the stability of the disordered local structure. The models are based on calculated correlations of softness and rearrangements. Without additional parameters, the models exhibit semiquantitative agreement with observed stress-strain curves and softness statistics for all systems studied. Moreover, the StEP models reveal that initial structure, the near-field effect of rearrangements on local structure, and rearrangement size, respectively, are responsible for the changes in ductility observed in the three systems. Thus, StEP models provide microscopic understanding of how strain localization depends on the interplay of structure, plasticity, and elasticity.

Correlation of plastic events with local structure in jammed packings across spatial dimensions.

In frictionless jammed packings, existing evidence suggests a picture in which localized physics dominates in low spatial dimensions, d = 2, 3, but quickly loses relevance as d rises, replaced by spatially extended mean-field behavior. For example, quasilocalized low-energy vibrational modes and low-coordination particles associated with deviation from mean-field behavior (rattlers and bucklers) all vanish rapidly with increasing d. These results suggest that localized rearrangements, which are associated with low-energy vibrational modes, correlated with local structure, and dominant in low dimensions, should give way in higher d to extended rearrangements uncorrelated with local structure. Here, we use machine learning to analyze simulations of jammed packings under athermal, quasistatic shear, identifying a local structural variable, softness, that correlates with rearrangements in dimensions d = 2 to d = 5. We find that softness­and even just the local coordination number Z­is essentially equally predictive of rearrangements in all d studied. This result provides direct evidence that local structure plays an important role in higher d, suggesting a modified picture for the dimensional cross-over to mean-field theory.

Transient learning degrees of freedom for introducing function in materials.

SignificanceMany protocols used in material design and training have a common theme: they introduce new degrees of freedom, often by relaxing away existing constraints, and then evolve these degrees of freedom based on a rule that leads the material to a desired state at which point these new degrees of freedom are frozen out. By creating a unifying framework for these protocols, we can now understand that some protocols work better than others because the choice of new degrees of freedom matters. For instance, introducing particle sizes as degrees of freedom to the minimization of a jammed particle packing can lead to a highly stable state, whereas particle stiffnesses do not have nearly the same impact.

Periodic training of creeping solids.

We consider disordered solids in which the microscopic elements can deform plastically in response to stresses on them. We show that by driving the system periodically, this plasticity can be exploited to train in desired elastic properties, both in the global moduli and in local "allosteric" interactions. Periodic driving can couple an applied "source" strain to a "target" strain over a path in the energy landscape. This coupling allows control of the system's response, even at large strains well into the nonlinear regime, where it can be difficult to achieve control simply by design.

Probing Gardner Physics in an Active Quasithermal Pressure-Controlled Granular System of Noncircular Particles.

To search for experimental signals of the Gardner crossover, an active quasithermal granular glass is constructed using a monolayer of air-fluidized star-shaped particles. The pressure of the system is controlled by adjusting the tension exerted on an enclosing boundary. Velocity distributions of the internal particles and the scaling of the pressure, density, effective temperature, and relaxation time are examined, demonstrating that the system has key features of a thermal system. Using a pressure-based quenching protocol that brings the system into deeper glassy states, signals of the Gardner crossover are detected via cage size and separation order parameters for both particle positions and orientations, offering experimental evidence of Gardner physics for a system of anisotropic quasithermal particles in a low spatial dimension.

Fragility in glassy liquids: A structural approach based on machine learning.

The rapid rise of viscosity or relaxation time upon supercooling is a universal hallmark of glassy liquids. The temperature dependence of viscosity, however, is quite nonuniversal for glassy liquids and is characterized by the system's "fragility," with liquids with nearly Arrhenius temperature-dependent relaxation times referred to as strong liquids and those with super-Arrhenius behavior referred to as fragile liquids. What makes some liquids strong and others fragile is still not well understood. Here, we explore this question in a family of harmonic spheres that range from extremely strong to extremely fragile, using "softness," a structural order parameter identified by machine learning to be highly correlated with dynamical rearrangements. We use a support vector machine to identify softness as the same linear combination of structural quantities across the entire family of liquids studied. We then use softness to identify the factors controlling fragility.

Limits of multifunctionality in tunable networks.

Nature is rife with networks that are functionally optimized to propagate inputs to perform specific tasks. Whether via genetic evolution or dynamic adaptation, many networks create functionality by locally tuning interactions between nodes. Here we explore this behavior in two contexts: strain propagation in mechanical networks and pressure redistribution in flow networks. By adding and removing links, we are able to optimize both types of networks to perform specific functions. We define a single function as a tuned response of a single "target" link when another, predetermined part of the network is activated. Using network structures generated via such optimization, we investigate how many simultaneous functions such networks can be programed to fulfill. We find that both flow and mechanical networks display qualitatively similar phase transitions in the number of targets that can be tuned, along with the same robust finite-size scaling behavior. We discuss how these properties can be understood in the context of constraint-satisfaction problems.

Hidden Topological Structure of Flow Network Functionality.

The ability to reroute and control flow is vital to the function of venation networks across a wide range of organisms. By modifying individual edges in these networks, either by adjusting edge conductances or creating and destroying edges, organisms robustly control the propagation of inputs to perform specific tasks. However, a fundamental disconnect exists between the structure and function: networks with different local architectures can perform the same functions. Here, we answer the question of how changes at the level of individual edges collectively create functionality at the scale of an entire network. Using persistent homology, we analyze networks tuned to perform complex tasks. We find that the responses of such networks encode a hidden topological structure composed of sectors of nearly uniform pressure. Although these sectors are not apparent in the underlying network structure, they correlate strongly with the tuned function. The connectivity of these sectors, rather than that of individual nodes, provides a quantitative relationship between structure and function in flow networks.

Elastoplasticity Mediates Dynamical Heterogeneity Below the Mode Coupling Temperature.

As liquids approach the glass transition temperature, dynamical heterogeneity emerges as a crucial universal feature of their behavior. Dynamic facilitation, where local motion triggers further motion nearby, plays a major role in this phenomenon. Here we show that long-ranged, elastically mediated facilitation appears below the mode coupling temperature, adding to the short-range component present at all temperatures. Our results suggest deep connections between the supercooled liquid and glass states, and pave the way for a deeper understanding of dynamical heterogeneity in glassy systems.

Quantifying the link between local structure and cellular rearrangements using information in models of biological tissues.

Machine learning techniques have been used to quantify the relationship between local structural features and variations in local dynamical activity in disordered glass-forming materials. To date these methods have been applied to an array of standard (Arrhenius and super-Arrhenius) glass formers, where work on "soft spots" indicates a connection between the linear vibrational response of a configuration and the energy barriers to non-linear deformations. Here we study the Voronoi model, which takes its inspiration from dense epithelial monolayers and which displays anomalous, sub-Arrhenius scaling of its dynamical relaxation time with decreasing temperature. Despite these differences, we find that the likelihood of rearrangements can nevertheless vary by several orders of magnitude within the model tissue and extract a local structural quantity, "softness," that accurately predicts the temperature dependence of the relaxation time. We use an information-theoretic measure to quantify the extent to which softness determines impending topological rearrangements; we find that softness captures nearly all of the information about rearrangements that is obtainable from structure, and that this information is large in the solid phase of the model and decreases rapidly as state variables are varied into the fluid phase.

Machine learning determination of atomic dynamics at grain boundaries.

In polycrystalline materials, grain boundaries are sites of enhanced atomic motion, but the complexity of the atomic structures within a grain boundary network makes it difficult to link the structure and atomic dynamics. Here, we use a machine learning technique to establish a connection between local structure and dynamics of these materials. Following previous work on bulk glassy materials, we define a purely structural quantity (softness) that captures the propensity of an atom to rearrange. This approach correctly identifies crystalline regions, stacking faults, and twin boundaries as having low likelihood of atomic rearrangements while finding a large variability within high-energy grain boundaries. As has been found in glasses, the probability that atoms of a given softness will rearrange is nearly Arrhenius. This indicates a well-defined energy barrier as well as a well-defined prefactor for the Arrhenius form for atoms of a given softness. The decrease in the prefactor for low-softness atoms indicates that variations in entropy exhibit a dominant influence on the atomic dynamics in grain boundaries.

Auxetic metamaterials from disordered networks.

Recent theoretical work suggests that systematic pruning of disordered networks consisting of nodes connected by springs can lead to materials that exhibit a host of unusual mechanical properties. In particular, global properties such as Poisson's ratio or local responses related to deformation can be precisely altered. Tunable mechanical responses would be useful in areas ranging from impact mitigation to robotics and, more generally, for creation of metamaterials with engineered properties. However, experimental attempts to create auxetic materials based on pruning-based theoretical ideas have not been successful. Here we introduce a more realistic model of the networks, which incorporates angle-bending forces and the appropriate experimental boundary conditions. A sequential pruning strategy of select bonds in this model is then devised and implemented that enables engineering of specific mechanical behaviors upon deformation, both in the linear and in the nonlinear regimes. In particular, it is shown that Poisson's ratio can be tuned to arbitrary values. The model and concepts discussed here are validated by preparing physical realizations of the networks designed in this manner, which are produced by laser cutting 2D sheets and are found to behave as predicted. Furthermore, by relying on optimization algorithms, we exploit the networks' susceptibility to tuning to design networks that possess a distribution of stiffer and more compliant bonds and whose auxetic behavior is even greater than that of homogeneous networks. Taken together, the findings reported here serve to establish that pruned networks represent a promising platform for the creation of unique mechanical metamaterials.

Disconnecting structure and dynamics in glassy thin films.

Nanometrically thin glassy films depart strikingly from the behavior of their bulk counterparts. We investigate whether the dynamical differences between a bulk and thin film polymeric glass former can be understood by differences in local microscopic structure. Machine learning methods have shown that local structure can serve as the foundation for successful, predictive models of particle rearrangement dynamics in bulk systems. By contrast, in thin glassy films, we find that particles at the center of the film and those near the surface are structurally indistinguishable despite exhibiting very different dynamics. Next, we show that structure-independent processes, already present in bulk systems and demonstrably different from simple facilitated dynamics, are crucial for understanding glassy dynamics in thin films. Our analysis suggests a picture of glassy dynamics in which two dynamical processes coexist, with relative strengths that depend on the distance from an interface. One of these processes depends on local structure and is unchanged throughout most of the film, while the other is purely Arrhenius, does not depend on local structure, and is strongly enhanced near the free surface of a film.

Relationship between local structure and relaxation in out-of-equilibrium glassy systems.

The dynamical glass transition is typically taken to be the temperature at which a glassy liquid is no longer able to equilibrate on experimental timescales. Consequently, the physical properties of these systems just above or below the dynamical glass transition, such as viscosity, can change by many orders of magnitude over long periods of time following external perturbation. During this progress toward equilibrium, glassy systems exhibit a history dependence that has complicated their study. In previous work, we bridged the gap between structure and dynamics in glassy liquids above their dynamical glass transition temperatures by introducing a scalar field called "softness," a quantity obtained using machine-learning methods. Softness is designed to capture the hidden patterns in relative particle positions that correlate strongly with dynamical rearrangements of particle positions. Here we show that the out-of-equilibrium behavior of a model glass-forming system can be understood in terms of softness. To do this we first demonstrate that the evolution of behavior following a temperature quench is a primarily structural phenomenon: The structure changes considerably, but the relationship between structure and dynamics remains invariant. We then show that the relaxation time can be robustly computed from structure as quantified by softness, with the same relation holding both in equilibrium and as the system ages. Together, these results show that the history dependence of the relaxation time in glasses requires knowledge only of the softness in addition to the usual state variables.

Designing allostery-inspired response in mechanical networks.

Recent advances in designing metamaterials have demonstrated that global mechanical properties of disordered spring networks can be tuned by selectively modifying only a small subset of bonds. Here, using a computationally efficient approach, we extend this idea to tune more general properties of networks. With nearly complete success, we are able to produce a strain between any two target nodes in a network in response to an applied source strain on any other pair of nodes by removing only ∼1% of the bonds. We are also able to control multiple pairs of target nodes, each with a different individual response, from a single source, and to tune multiple independent source/target responses simultaneously into a network. We have fabricated physical networks in macroscopic 2D and 3D systems that exhibit these responses. This work is inspired by the long-range coupled conformational changes that constitute allosteric function in proteins. The fact that allostery is a common means for regulation in biological molecules suggests that it is a relatively easy property to develop through evolution. In analogy, our results show that long-range coupled mechanical responses are similarly easy to achieve in disordered networks.

Scaling ansatz for the jamming transition.

We propose a Widom-like scaling ansatz for the critical jamming transition. Our ansatz for the elastic energy shows that the scaling of the energy, compressive strain, shear strain, system size, pressure, shear stress, bulk modulus, and shear modulus are all related to each other via scaling relations, with only three independent scaling exponents. We extract the values of these exponents from already known numerical or theoretical results, and we numerically verify the resulting predictions of the scaling theory for the energy and residual shear stress. We also derive a scaling relation between pressure and residual shear stress that yields insight into why the shear and bulk moduli scale differently. Our theory shows that the jamming transition exhibits an emergent scale invariance, setting the stage for the potential development of a renormalization group theory for jamming.

Mechanical signaling coordinates the embryonic heartbeat.

In the beating heart, cardiac myocytes (CMs) contract in a coordinated fashion, generating contractile wave fronts that propagate through the heart with each beat. Coordinating this wave front requires fast and robust signaling mechanisms between CMs. The primary signaling mechanism has long been identified as electrical: gap junctions conduct ions between CMs, triggering membrane depolarization, intracellular calcium release, and actomyosin contraction. In contrast, we propose here that, in the early embryonic heart tube, the signaling mechanism coordinating beats is mechanical rather than electrical. We present a simple biophysical model in which CMs are mechanically excitable inclusions embedded within the extracellular matrix (ECM), modeled as an elastic-fluid biphasic material. Our model predicts strong stiffness dependence in both the heartbeat velocity and strain in isolated hearts, as well as the strain for a hydrogel-cultured CM, in quantitative agreement with recent experiments. We challenge our model with experiments disrupting electrical conduction by perfusing intact adult and embryonic hearts with a gap junction blocker, ß-glycyrrhetinic acid (BGA). We find this treatment causes rapid failure in adult hearts but not embryonic hearts-consistent with our hypothesis. Last, our model predicts a minimum matrix stiffness necessary to propagate a mechanically coordinated wave front. The predicted value is in accord with our stiffness measurements at the onset of beating, suggesting that mechanical signaling may initiate the very first heartbeats.

Two Diverging Length Scales in the Structure of Jammed Packings.

At densities higher than the jamming transition for athermal, frictionless repulsive spheres we find two distinct length scales, both of which diverge as a power law as the transition is approached. The first, ξ_{Z}, is associated with the two-point correlation function for the number of contacts on two particles as a function of the particle separation. The second, ξ_{f}, is associated with contact-number fluctuations in subsystems of different sizes. On scales below ξ_{f}, the fluctuations are highly suppressed, similar to the phenomenon of hyperuniformity usually associated with density fluctuations. The exponents for the divergence of ξ_{Z} and ξ_{f} are different and appear to be different in two and three dimensions.