Amaury Hayat

Guillaume Lample, Marie-Anne Lachaux, Thibaut Lavril et al.

We propose an online training procedure for a transformer-based automated theorem prover. Our approach leverages a new search algorithm, HyperTree Proof Search (HTPS), inspired by the recent success of AlphaZero. Our model learns from previous proof searches through online training, allowing it to generalize to domains far from the training distribution. We report detailed ablations of our pipeline's main components by studying performance on three environments of increasing complexity. In particular, we show that with HTPS alone, a model trained on annotated proofs manages to prove 65.4% of a held-out set of Metamath theorems, significantly outperforming the previous state of the art of 56.5% by GPT-f. Online training on these unproved theorems increases accuracy to 82.6%. With a similar computational budget, we improve the state of the art on the Lean-based miniF2F-curriculum dataset from 31% to 42% proving accuracy.

Fabian Gloeckle, Mantas Baksys, Darius Feher et al.

Recent progress in large language models (LLMs) has advanced automatic code generation and formal theorem proving, yet software verification has not seen the same improvement. To address this gap, we propose WybeCoder, an agentic code verification framework that enables prove-as-you-generate development where code, invariants, and proofs co-evolve. It builds on a recent framework that combines automatic verification condition generation and SMT solvers with interactive proofs in Lean. To enable systematic evaluation, we translate two benchmarks for functional verification in Lean, Verina and Clever, to equivalent imperative code specifications. On complex algorithms such as Heapsort, we observe consistent performance improvements by scaling our approach, synthesizing dozens of valid invariants and dispatching of dozens of subgoals, resulting in hundreds of lines of verified code, overcoming plateaus reported in previous works. Our best system solves 74% of Verina tasks and 62% of Clever tasks at moderate compute budgets, significantly surpassing previous evaluations and paving a path to automated construction of large-scale datasets of verified imperative code.

Georges Bastin, Jean-Michel Coron, Amaury Hayat

We study a system of several one-dimensional scalar conservation laws coupled through boundary feedback conditions that combine physical boundary constraints with static feedback control laws. Our first contribution establishes the well-posedness of the system in the space of $L^{\infty}$ entropy solutions. Our second contribution provides a set of sufficient dissipative conditions on the boundary coupling that ensure global exponential stability in the $L^1$ and $L^\infty$ norms.

Antoine Peyronnet, Fabian Gloeckle, Amaury Hayat

We present a new approach for benchmarking Large Language Model (LLM) capabilities on research-level mathematics. Existing benchmarks largely rely on static, hand-curated sets of contest or textbook-style problems as proxies for mathematical research. Instead, we establish an updatable benchmark evaluating models directly on the latest research results in mathematics. This consists of an automatic pipeline that extracts lemmas from arXiv and rewrites them into self-contained statements by making all assumptions and required definitions explicit. It results in a benchmark that can be updated regularly with new problems taken directly from human mathematical research, while previous instances can be used for training without compromising future evaluations. We benchmark current state-of-the-art LLMs, which obtain around 10-15$\%$ accuracy in theorem proving (pass@1) depending on the model, showing that there is currently a large margin of progression for LLMs to reach human-level proving capabilities in a research context.

Nail Baloul, Amaury Hayat, Thibault Liard et al.

We propose an efficient method for reconstructing traffic density with low penetration rate of probe vehicles. Specifically, we rely on measuring only the initial and final positions of a small number of cars which are generated using microscopic dynamical systems. We then implement a machine learning algorithm from scratch to reconstruct the approximate traffic density. This approach leverages learning techniques to improve the accuracy of density reconstruction despite constraints in available data. For the sake of consistency, we will prove that, if only using data from dynamical systems, the approximate density predicted by our learned-based model converges to a well-known macroscopic traffic flow model when the number of vehicles approaches infinity.

François Charton, Amaury Hayat, Sean T. McQuade et al.

We show that deep learning models, and especially architectures like the Transformer, originally intended for natural language, can be trained on randomly generated datasets to predict to very high accuracy both the qualitative and quantitative features of metabolic networks. Using standard mathematical techniques, we create large sets (40 million elements) of random networks that can be used to train our models. These trained models can predict network equilibrium on random graphs in more than 99% of cases. They can also generalize to graphs with different structure than those encountered at training. Finally, they can predict almost perfectly the equilibria of a small set of known biological networks. Our approach is both very economical in experimental data and uses only small and shallow deep-learning model, far from the large architectures commonly used in machine translation. Such results pave the way for larger use of deep learning models for problems related to biological networks in key areas such as quantitative systems pharmacology, systems biology, and synthetic biology.

François Charton, Amaury Hayat, Guillaume Lample

Using transformers over large generated datasets, we train models to learn mathematical properties of differential systems, such as local stability, behavior at infinity and controllability. We achieve near perfect prediction of qualitative characteristics, and good approximations of numerical features of the system. This demonstrates that neural networks can learn to perform complex computations, grounded in advanced theory, from examples, without built-in mathematical knowledge.