Using Machine Learning To Study The String Landscape

Is fundamental physics unified into a single theory governing all known phenomena, or are we forced to accept a fractured state of affairs where different phenomena are addressed by different theories?

This question has long been of first importance to theoretical physicists. Einstein, for example, spent many of his later years in search for a unified theory, with little success. Despite his brilliance, the deck was stacked against him, as certain aspects of fundamental physics such as the strong and weak nuclear forces were only just being discovered at the end of his life.

Today we have a more complete picture of the interactions of elementary particles and also a strong sense of what is difficult in the search for a unified theory: combining general relativity, Einstein’s theory of gravity, with quantum mechanics. The search for a unified theory is, therefore, a search for a quantum theory of gravity that has the ability to recover known phenomena in particle physics and cosmology, including the entire standard model of particle physics that has been tested for decades at particle accelerators such as the Large Hadron Collider. This search continues today.

String is a quantum theory of gravity that is perhaps the most promising candidate for a unified theory of physics. It satisfies a number of non-trivial necessary conditions that must be satisfied by any unified theory, including recovering general relativity at long distances and naturally giving rise to the building blocks of realistic cosmological and particle sectors. For these reasons string theory has been a primary focus of theoretical high energy physicists since an important breakthrough in 1984. In addition to continued progress toward unification, string theory has also spawned new subfields in physics and mathematics.

However, its extra dimensions of space must be wrapped up in a “compactification” in order to recover the three spatial dimensions that we observe, and there are many possible ways to do so. There are also many possible configurations of generalizations of electromagnetic fluxes in the extra dimensions. Together, these lead to a large “landscape” of solutions, known as vacua, and the different solutions realize many different incarnations of particle physics and cosmology. Taming the landscape is, therefore, a central problem in theoretical physics, and is critical to making progress in understanding unification in string theory.

In recent work entitled “Machine Learning in the String Landscape” with Jonathan Carifio and Dmitri Krioukov, we proposed treating the landscape as what it clearly is: a big data problem. In fact, the data that arise in string theory may be some of the largest in science. For example, in a region of the landscape known as type IIb, it was originally estimated that there are 10⁵⁰⁰ vacua arising from different choices of fluxes, which has recently exploded to 10^272,000.  It is now known that there are at least 10⁷⁵⁵ geometries in which fluxes can be turned on, which is an exact lower bound, and recent estimates imply there are at least 10³⁰⁰⁰ geometries. Numbers of this magnitude are extremely difficult to handle, a problem which is further exacerbated by the computational complexity of the landscape. It is clear that sophisticated techniques are required.

Such problems may be tailor-made for applying modern techniques in data science. Over the last few years, application of data science techniques to known problems has changed many fields. In one well-known example, with under two days of self-training and no human knowledge input into the training process, DeepMind’s program AlphaGo Zero was able to master the game of Go, achieving a level of play beyond that of AlphaGo Lee, which itself defeated the world champion Lee Sedol in 2016. In chess, AlphaZero outperforms the previously strongest program Stockfish with only four hours of training, even though Stockfish is itself stronger than any living grandmaster. Beyond the realm of board games, data science has led to veritable revolutions in genetics, oceanography, climate science, and many other fields.

Can techniques in data science make super-human progress in the string landscape, bringing us closer to unifying fundamental physics? It is far too early to tell.

Nevertheless, a number of string theorists are intrigued by the possibility. Multiple works over the last six months have brought data science techniques into diverse problems in string theory, particularly those that are relevant for constructing geometries and counting numbers of particles. When the sizes of datasets are small enough, standard machine learning techniques may be applied. For example, we found that supervised machine learning accurately estimates the number of a certain type of extra dimensional geometry known as a toric weak-Fano threefold.

But what should be done when the datasets are so large that they cannot be read into a computer? At first thought, such problems seem intractable, and in general, they may be. However, in one case we found a way out. First, we randomly sampled a large ensemble of geometries and used machine learning to train a model to predict whether a certain feature of particle physics arises in a geometry. The physical model is not yet complete, but the feature is related to so-called E₆ grand unification, which is one candidate for unifying the strong and weak nuclear force together with electromagnetism into one force. This feature was predicted by the machine-learned logistic regression model with 98% accuracy, which also shed light on the variables in the data that determined whether or not the feature was present. The determination of one critical variable by the model led to a concrete mathematical conjecture that we were able to prove rigorously. By using machine learning to generate conjectures, we were able to take the result of a data analysis of 10 million random samples and turn it into a rigorous result about properties of 10⁷⁵⁵ geometries. More broadly, using machine learning for conjecture generation could lead to important rigorous results in the extremely large data sets in string theory.

New and upcoming results were recently reported at the Workshop on Data Science and String Theory at Northeastern University,  held from November 30 – December 2, 2017. Over 50 participants from all over the world gathered to discuss the application of data science to string theory, bringing together academia and industry for perhaps the first time at a string theory conference. New results were announced, including the introduction of data science techniques that are new or newer to string theory, such as network science, topological data analysis, and deep reinforcement learning. Additional meetings of this type are being planned for 2018.

We do not know where this new direction will lead, but given the promise of string theory, the enormity of its datasets, and the importance of unification, it promises to be an interesting avenue of research in the coming years.

The study, Machine learning in the string landscape was recently published in the Journal of High Energy Physics.