Researchers at Meta’s Fundamental AI Research (FAIR) group have trained an artificial-intelligence system that can propose Lyapunov functions, the mathematical “energy maps” used to prove whether a dynamic system will stay stable over time. For more than 130 years, mathematicians have lacked a general way to find these functions, leaving problems such as the famous three-body question beyond analytical reach.
Instead of hand-coding rules, the team generated millions of random examples where both the dynamical system and a valid Lyapunov function are known. A large sequence-to-sequence transformer was then taught to “translate” a new system directly into a candidate stability function. In benchmark tests the model outperformed specialist algorithms and human experts, and even discovered Lyapunov functions for non-polynomial systems that previously had none.
The breakthrough matters because Lyapunov proofs underpin safety guarantees everywhere from spacecraft trajectories and power-grid control to robotic balance. An automated tool that can suggest workable functions gives mathematicians and engineers a starting point they can rigorously verify. Meta emphasises that human checking is still essential: the AI proposes explicit formulas, but formal proof steps must be confirmed manually or by computer algebra.
Beyond this result, the work is a demonstration that language-model architectures trained on synthetic data can tackle open mathematical questions, hinting at a new workflow where machines generate plausible solutions and humans supply the final proof.
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