Statistical Patterns in the Equations of Physics and the Emergence of a Meta-Law of Nature

Physics, as a fundamental science, aims to understand the laws of Nature and describe them in mathematical equations. While the physical reality manifests itself in a wide range of phenomena with varying levels of complexity, the equations that describe them display certain statistical regularities and patterns, which we begin to explore here. By drawing inspiration from linguistics, where Zipf's law states that the frequency of any word in a large corpus of text is roughly inversely proportional to its rank in the frequency table, we investigate whether similar patterns for the distribution of operators emerge in the equations of physics. We analyse three corpora of formulae and find, using sophisticated implicit-likelihood methods, that the frequency of operators as a function of their rank in the frequency table is best described by an exponential law with a stable exponent, in contrast with Zipf's inverse power-law. Understanding the underlying reasons behind this statistical pattern may shed light on Nature's modus operandi or reveal recurrent patterns in physicists' attempts to formalise the laws of Nature. It may also provide crucial input for symbolic regression, potentially augmenting language models to generate symbolic models for physical phenomena. By pioneering the study of statistical regularities in the equations of physics, our results open the door for a meta-law of Nature, a (probabilistic) law that all physical laws obey.

Decoding Nature with Nature's Tools: Heterotic Line Bundle Models of Particle Physics with Genetic Algorithms and Quantum Annealing

The string theory landscape may include a multitude of ultraviolet embeddings of the Standard Model, but identifying these has proven difficult due to the enormous number of available string compactifications. Genetic Algorithms (GAs) represent a powerful class of discrete optimisation techniques that can efficiently deal with the immensity of the string landscape, especially when enhanced with input from quantum annealers. In this letter we focus on geometric compactifications of the $E_8\times E_8$ heterotic string theory compactified on smooth Calabi-Yau threefolds with Abelian bundles. We make use of analytic formulae for bundle-valued cohomology to impose the entire range of spectrum requirements, something that has not been possible so far. For manifolds with a relatively low number of Kahler parameters we compare the GA search results with results from previous systematic scans, showing that GAs can find nearly all the viable solutions while visiting only a tiny fraction of the solution space. Moreover, we carry out GA searches on manifolds with a larger numbers of Kahler parameters where systematic searches are not feasible.

Intelligent Explorations of the String Theory Landscape

The goal of identifying the Standard Model of particle physics and its extensions within string theory has been one of the principal driving forces in string phenomenology. Recently, the incorporation of artificial intelligence in string theory and certain theoretical advancements have brought to light unexpected solutions to mathematical hurdles that have so far hindered progress in this direction. In this review we focus on model building efforts in the context of the $E_8\times E_8$ heterotic string compactified on smooth Calabi-Yau threefolds and discuss several areas in which machine learning is expected to make a difference.

Heterotic String Model Building with Monad Bundles and Reinforcement Learning

We use reinforcement learning as a means of constructing string compactifications with prescribed properties. Specifically, we study heterotic SO(10) GUT models on Calabi-Yau three-folds with monad bundles, in search of phenomenologically promising examples. Due to the vast number of bundles and the sparseness of viable choices, methods based on systematic scanning are not suitable for this class of models. By focusing on two specific manifolds with Picard numbers two and three, we show that reinforcement learning can be used successfully to explore monad bundles. Training can be accomplished with minimal computing resources and leads to highly efficient policy networks. They produce phenomenologically promising states for nearly 100% of episodes and within a small number of steps. In this way, hundreds of new candidate standard models are found.