Authors: Jonathan J. Heckman, Shani Meynet, Alessandro Mininno, Gary Shiu
Organizations: Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104, USA · Department of Physics, University of Wisconsin–Madison, 1150 University Avenue, Madison, WI 53706, USA
Dualities play an important role in establishing both microscopic and emergent phenomena in a wide range of physical systems. In practice, though, it can often be computationally challenging to establish when two systems are dual, even when all of the "rules of the game" are well-known. Said differently, when confronted with two systems, how can one efficiently establish that they are in fact dual? In this paper we use machine learning methods to address this question for Seiberg dualities of supersymmetric quiver gauge theories. Mathematically, this involves establishing mutations of quivers, which is in turn a variation on the theme of "learning to unknot". On the one hand, this leads us to a practical tool for establishing the computational complexity of different dualities. On the other hand, it also allows us to study how different network architectures learn how to trace Seiberg dualities. We find that for quivers with a modest number of quiver nodes (of order 10), different network architectures consisting of transformers and multi-layer perceptrons tend to outperform deterministic algorithms. Supplementing the network by well-established pathfinder algorithms (essentially "Google Maps for quivers") leads to an additional improvement in the efficiency and accuracy of the search strategy. We anticipate that this class of questions can serve as a useful benchmark for frontier AI models applied to theoretical physics.
We introduce dualGNN', an autoregressive message-passing GNN for sampling fine, regular triangulations of lattice polytopes. dualGNN operates on a generalization of the dual graph of a triangulation, with edges labeled by signed circuits' -- combinatorial invariants from the theory of oriented matroids. We show that these circuits are necessary and sufficient to determine a triangulation's regularity from the graph, provided certain magnitude information is retained. The model is independent of the polytope's point count and invariant under its orientation-preserving symmetries (SL(d,Z)⋉Zd), and our masking procedure further guarantees that every rollout produces a fine triangulation (in 2D). On unseen polygons with Npts≤40, dualGNN is the only sampler we tested that is consistent with uniform sampling across all our diagnostics (KL divergence from uniformity, collision counts, and sample autocorrelation). The model is small (∼92k parameters) and trains in ∼7.5 hours on a single consumer GPU. We apply dualGNN to string theory, sampling Calabi-Yau threefolds uniformly at h1,1=86; we also sample CYs at h1,1=128, observing no deviations from uniformity, but our diagnostics are weaker here. Code, training scripts, and pretrained models are available at https://github.com/natemacfadden/dualGNN (pip install dualgnn), and dualGNN is integrated into CYTools.
Algorithmic reasoning -- the ability to perform step-by-step logical inference -- is a synthetic benchmark for evaluating multi-step reasoning abilities, designed for graph neural networks and also for transformer models. Prior work has evaluated reasoning for executing a single algorithmic task, whereas a more desirable objective is to perform multiple algorithmic reasoning tasks simultaneously. We start by noting that this is inherently difficult due to differences arising from the execution traces of the algorithms (such as depth- vs. breadth-first search), which cause interference when they are trained together. In this paper, we introduce {branching neural networks}, a new architecture for multitask algorithmic reasoning. The main idea is to search for a recursive tree-structured partition of n algorithmic tasks into a k-ary tree (divided into L layers). Naive search requires O(knL) complexity; we develop an algorithm that reduces this to O(nL) by solving a convex relaxation at each layer to approximate an optimal partition. Our approach clusters these tasks using gradient-based affinity and can be used on top of any base model. We validate our approach on algorithmic reasoning benchmarks and their extensions with text descriptions. We show that gradient-based affinity scores help estimate true performance with less than 5% error, measured across eight different architectures with up to 34 billion parameters. On the CLRS benchmark, our approach outperforms existing graph neural networks by 3.7% and baselines by 1.2%, while reducing runtime by 48% and memory usage by 26%. The learned branching structure shows a hierarchical clustering of related algorithms. On three text-based graph reasoning benchmarks, our approach improves over baseline methods by 3.2%. Finally, we validate our approach for overlapping community detection.
We study a graph classification problem involving over 20 million graphs, arising from high-order perturbative computations of correlators in planar N=4 super-Yang--Mills, a model closely related to the theory of the strong nuclear force. We benchmark graph neural networks, including graph transformers, achieving robust generalization to larger graphs with up to 99.996% ROC AUC. Then, we analyze how the models can be used to gain a computational speedup compared to the traditional graphical bootstrap algorithm, through shrinking the redundant data by up to 85.5% at the level of denominator graphs. Finally, we study the embeddings of the models to investigate their interpretability.