Towards understanding glasses with graph neural networks

Underneath a microscope, a pane of window glass doesn’t appear to be a set of orderly molecules, as a crystal would, however fairly a jumble with no discernable construction. Glass is made by beginning with a glowing combination of high-temperature melted sand and minerals. As soon as cooled, its viscosity (a measure of the friction within the fluid) will increase a trillion-fold, and it turns into a strong, resisting rigidity from stretching or pulling. But the molecules within the glass stay in a seemingly disordered state, very like the unique molten liquid – virtually as if the disordered liquid state had been flash-frozen in place. The glass transition, then, first seems to be a dramatic arrest within the motion of the glass molecules. Whether or not this course of corresponds to a structural section transition (as in water freezing, or the superconducting transition) is a significant open query within the subject. Understanding the character of the dynamics of glass is prime to understanding how the atomic-scale properties outline the seen options of many strong supplies.  Within the phrases of the just lately deceased Nobel Prize laureate Philip W. Anderson, whose pioneering work formed the sector of solid-state physics:

The deepest and most fascinating unsolved drawback in strong state principle might be the idea of the character of glass and the glass transition.
Philip W. Anderson

Determine 1: A liquid, when cooled too shortly previous its crystallisation level, turns right into a supercooled liquid which, upon additional cooling, turns right into a disordered, amorphous glass. If cooled slowly sufficient, it might as a substitute remodel into an ordered crystal.

The sensible implications of modelling glass

The glass transition is a ubiquitous phenomenon which manifests in additional than window (silica) glasses. For example, when ironing, polymers in a material are heated, turn out to be cellular, after which oriented by the load of the iron. Extra broadly, an identical and associated transition, the jamming transition, could be present in colloidal suspensions (equivalent to ice cream), granular supplies (equivalent to a static pile of sand), and in addition organic techniques (e.g., for modelling cell migration throughout embryonic growth) in addition to social behaviours (for example visitors jams). These techniques all function below native constraints the place the place of some parts inhibits the movement of others (termed frustration). Their dynamics are advanced and cooperative, taking the type of large-scale, collective rearrangements which propagate via area in a heterogeneous method. Glasses are thought of to be archetypal of those sorts of advanced techniques, and so higher understanding them can have implications throughout many analysis areas. This understanding may yield sensible advantages – for instance, creating supplies which have a extra secure glass construction, as a substitute of a crystalline one, would enable them to dissolve shortly, which may result in new drug supply strategies.  Understanding the glass transition could lead to different functions of disordered supplies, in fields as various as biorenewable polymers and meals processing. The research of glasses has additionally already led to insights in apparently very totally different domains equivalent to constraint satisfaction issues in pc science and, extra just lately, the coaching dynamics of under-parameterized neural networks.

A deeper understanding of glasses could result in sensible advances sooner or later, however their mysterious properties additionally elevate many basic analysis questions. Although people have been making silica glasses for not less than 4 thousand years, they continue to be enigmatic to scientists: there are various unknowns in regards to the underlying bodily correlates of, for instance, the trillion-fold improve in viscosity that occurs over the cooling course of. Our curiosity on this subject was additionally motivated by the truth that glasses are additionally a superb testbed for making use of trendy machine studying strategies to bodily issues: they’re straightforward to simulate, and straightforward to enter to particle-based machine studying fashions. Crucially, we will then go in and look at these fashions to know what they’ve realized in regards to the system, to realize deeper qualitative insights in regards to the nature of glass, and the structural portions which underpin its mysterious dynamical qualities. Our new work, revealed in Nature Physics, may assist us achieve an understanding of the structural modifications that will happen close to the glass transition. Extra virtually, this analysis may result in insights in regards to the mechanical constraints of glasses (e.g., the place a glass will break).

Leveraging graph neural networks to mannequin glassy dynamics

Glasses could be modelled as particles interacting by way of a short-range repulsive potential which primarily prevents particles from getting too shut to one another. This potential is relational (solely pairs of particles work together) and native (solely close by particles work together with one another), which suggests {that a} mannequin that respects this native and relational construction ought to be efficient. In different phrases, given the system is underpinned by a graph-like construction, we reasoned it will be greatest modeled by a graph structured community, and got down to apply Graph Neural Networks to foretell bodily elements of a glass.

We first created an enter graph the place the nodes signify particles, and edges signify interactions between particles, and are labelled with their relative distance.  A particle was linked to its neighboring particles inside a sure radius (on this case, 2 particle diameters). We then skilled a neural community, described beneath, to foretell a single actual quantity for every node of the graph. This prediction was in the end regressed in the direction of the mobilities of particles obtained from pc simulations of glasses. Mobility is a measure of how a lot a particle usually strikes (extra technically, this corresponds to the typical distance travelled when averaging over preliminary velocities).

Determine 2: Mannequin structure. a) From the 3D inputs, nodes at distance lower than 2 are linked to type a graph. After processing, the community predicts mobilities (represented by totally different colors) for every particle. b) The graph community’s core first updates edges primarily based on their earlier embedding and people of their adjoining nodes, after which nodes primarily based on their earlier embeddings and people of incoming edges. c) The graph community consists of an encoder, a number of functions of the core, adopted by a decoder. Every software of the core will increase the shell of particles contributing to a given particle’s prediction, right here proven in color for the central particle (darkish blue).

Our community structure was a typical graph community structure, consisting of a number of neural networks. We first embedded the node and edge labels in a high-dimensional vector-space utilizing two encoder networks (we used normal multi-layer perceptrons). Subsequent, we iteratively up to date the embedded node and edge labels utilizing two replace networks visualized in Fig. 2b. At first, every edge up to date primarily based on its earlier embedding and the embeddings of the 2 nodes it linked to. In spite of everything edges had been up to date in parallel utilizing the identical community, the nodes had been additionally up to date primarily based on the sum of their neighboring edge embeddings and their earlier embeddings, utilizing a second community. We repeated this process a number of instances (usually 7), permitting native data to propagate all through the graph, as proven in Fig. 2c. Lastly, we extracted the mobility for every particle from the ultimate embeddings of the corresponding node utilizing a decoder community. The ensuing community has all of the required properties: it’s inherently relational, it’s invariant below permutation of the nodes and edges of the graph, and it updates embedding in a approach that may be a composition of native operations. The community parameter coaching was completed by way of stochastic gradient descent.

To check the total dynamical evolution of glasses, we constructed a number of datasets similar to predictions of mobilities on totally different time horizons and for various temperatures. We be aware that every particle can have collided a number of hundreds of instances over these timescales. Thus, the community should discover a method to coarsely signify the long-term dynamics of the system.

Connecting the community’s prediction with physics

After making use of graph networks to the three dimensional glasses that we simulated, we discovered that they strongly outperformed present fashions, starting from normal physics-inspired baselines to state-of-the-art machine studying fashions. Evaluating the anticipated mobilities (color gradients, Determine 3) with the bottom fact simulation (dots, Determine 3), we see that the settlement is extraordinarily good on brief instances and stays properly matched as much as the relief time of the glass. Taking a look at a glass over the timescale of its rest time – for precise glass, this might be hundreds of years – is like a liquid over a couple of picosecond (10-12): the relief time is loosely when particles have collided sufficient to begin dropping details about their preliminary place. In numbers, the correlation between our prediction and the simulation’s floor fact is 96% for very brief timescales, and stays excessive at 64% for the relief time of the glass (an enchancment of 40% in comparison with the earlier state-of-the-art).

3D prediction over brief timescales

Determine 3a: GNN-predicted mobilities (colored from least cellular in blue to most cellular in purple) in comparison with the place of probably the most cellular particles within the simulation (dots) in a slice of our three-dimensional field. Higher efficiency corresponds to higher alignment of purple areas and dots. This panel corresponds to a prediction over a brief timescale: a regime by which our community attains a really robust efficiency.

3D prediction over lengthy timescales

Determine 3b. On this panel, similar to a timescale 28,000 instances longer than the highest panel, particles within the glass have began to diffuse. The dynamics are heterogeneous – particle mobilities are correlated regionally, however heterogeneous at macroscopic scales – but our community nonetheless makes predictions in settlement with the bottom fact simulation.

We don’t need to merely mannequin glass, nonetheless: we need to perceive it.  We subsequently explored what components had been necessary to our mannequin’s success with a purpose to infer what properties are necessary within the underlying system. A central unsolved query within the dynamics of glass is how particles affect each other as a perform of distance, and the way this evolves over time. We investigated this by designing an experiment leveraging the particular structure of the graph community. Recall that repeated functions of the sting and node updates outline shells of particles round any given particle: the primary shell consists of all particles one step away from this “marked” particle, the second shell consists of all particles one step away from the primary shell, and so forth (see the totally different shades of blue on Determine 2c). By measuring the sensitivity of the prediction that the community makes for the central particle when the n-th shell is modified, we will measure how massive an space the community makes use of to extract its prediction, which offers an estimate of the gap over which particles affect one another within the bodily system.

Determine 4: Ablation experiment. On the left experiment, all particles past the primary shell round one central particle are eliminated. On the precise experiment, the enter is perturbed by rising the gap between the primary and second shells of particles.

We discovered that when predicting what occurs within the close to future or within the liquid section, drastic modifications of the third shell (for example, eradicating it altogether, Determine 4, left) did not modify the prediction that the community would make for the marked particle. Alternatively, when making predictions at low temperature and within the far future, after the glass begins to calm down, even tiny perturbations (Determine 4, proper) of the 5-th shell have an effect on the prediction for the marked particle. These findings are in keeping with a bodily image the place a correlation size (a measure of the gap over which particles affect one another) grows upon approaching the glass transition. The definition and research of correlation lengths is a cornerstone of the research of section transition in physics, and one that’s nonetheless an open level of debate when finding out glasses. Whereas this “machine realized” correlation size can’t be straight reworked right into a bodily measurable amount, it offers compelling proof that rising spatial correlations are current within the system upon approaching the glass transition, and that our community has realized to extract them.


Our outcomes present that graph networks represent a robust device to foretell the long run dynamics of glassy techniques, leveraging the construction hidden in a neighborhood neighborhood of particles. We anticipate our method to be helpful for predicting different bodily portions of curiosity in glasses, and hope that it’ll result in extra insights for glassy system theorists – we’re open-sourcing our fashions and skilled networks to help this effort. Extra usually, graph networks are a flexible device which might be being utilized to many different bodily techniques that include many-body interactions, in contexts together with visitors, crowd simulations, and cosmology. The community evaluation strategies used right here additionally yield a deeper understanding in different fields: graph networks could not solely assist us make higher predictions for a variety of techniques, however point out what bodily correlates are necessary for modeling them – on this work, how interactions between native particles in a glassy materials evolve over time.

We consider that our outcomes advocate utilizing structured fashions when making use of machine studying to the bodily sciences; in our case, the flexibility to analyse the interior workings of a neural community indicated that it had found a amount that correlates with an elusive bodily amount. This demonstrates that machine studying can be utilized not solely to make quantitative predictions, but in addition to realize qualitative understanding of bodily techniques. This might imply that machine studying techniques may have the ability to ultimately help researchers in deriving basic bodily theories, in the end serving to to reinforce, fairly than change, human understanding.

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