Global Lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers

Despite their spectacular progress, language models still struggle on complex reasoning tasks, such as advanced mathematics. We consider a long-standing open problem in mathematics: discovering a Lyapunov function that ensures the global stability of a dynamical system. This problem has no known general solution, and algorithmic solvers only exist for some small polynomial systems. We propose a new method for generating synthetic training samples from random solutions, and show that sequence-to-sequence transformers trained on such datasets perform better than algorithmic solvers and humans on polynomial systems, and can discover new Lyapunov functions for non-polynomial systems.

Teaching Transformers Modular Arithmetic at Scale

Modular addition is, on its face, a simple operation: given $N$ elements in $\mathbb{Z}_q$, compute their sum modulo $q$. Yet, scalable machine learning solutions to this problem remain elusive: prior work trains ML models that sum $N \le 6$ elements mod $q \le 1000$. Promising applications of ML models for cryptanalysis-which often involve modular arithmetic with large $N$ and $q$-motivate reconsideration of this problem. This work proposes three changes to the modular addition model training pipeline: more diverse training data, an angular embedding, and a custom loss function. With these changes, we demonstrate success with our approach for $N = 256, q = 3329$, a case which is interesting for cryptographic applications, and a significant increase in $N$ and $q$ over prior work. These techniques also generalize to other modular arithmetic problems, motivating future work.