Scaling Mathematical Proof Supervision from Structured Textbooks
Youqing Fang, Zhouqi Hua, Yinhao Tang, Yuzhe Gu, Lingkai Kong, Bin Liu, Weiming Zhang, Kai Chen 等 10 位作者
AI 速览：将2,118本大学数学教科书转化为约38万条保留章节顺序与命题-证明结构的证明监督记录，并研究课程学习与规模规律。
Mathematical supervision for language models is typically flattened into isolated problem-solution pairs, discarding the curricular organization and evidential structure of the original sources. We introduce TOME-Math (Textbook-Oriented Mathematical Evidence), a large-scale corpus that converts 2,118 textbooks spanning the core areas of university-level mathematics subjects into approximately 380K structured proof-supervision records while preserving chapter order, proposition-proof links, dependency references, and source provenance. Leveraging these structural signals, we explore curriculum learning across multiple mathematical subjects and characterize scaling patterns with supervision volume and source breadth. At full scale, fine-tuning on TOME-Math raises the unweighted mean score across our held-out evaluations and established public benchmarks by 16.7 percentage points over the base model, while exceeding all three public instruction-data baselines evaluated in aggregate. Our findings highlight structured textbooks as both a scalable source of mathematical supervision and an empirical framework for studying the impact of data organization on mathematical reasoning. We will release the corpus to facilitate further research on structured mathematical supervision.
Large language models (LLMs) draw on parametric knowledge and retrieval-augmented generation (RAG) to solve knowledge-intensive tasks. Yet problem solving in unfamiliar domains calls for tightly coupled knowledge acquisition and reasoning, beyond factual retrieval and aggregation or learning solely from a preassembled context. We study self-learning -- autonomously retrieving, studying, and applying knowledge from long-form reference materials on the fly. To investigate this capability, we formulate a textbook-based mathematical problem-solving task. LLMs must navigate textbook exposition, learn and apply relevant principles, and construct proofs for exercises, with reference solutions withheld. We introduce a BookNavigator interface to support this human-inspired study process and incentivize self-learning through reinforcement learning (RL) with fine-grained rewards. Experiments show that our method improves performance across in-domain and out-of-domain textbooks and subjects, raising Intern-S2-Preview-35B's overall proof accuracy from 51.5% to 79.5% and outperforming RL-trained textbook-free and fixed-retrieval baselines. These gains extend to complex long-context benchmarks involving different task types, with improvements over the base model of 1.7 and 2.3 percentage points on BrowseComp-Plus and CL-bench, respectively. Together, these findings suggest that RL can foster a domain-agnostic self-learning strategy, highlighting the potential of self-learning from long-form corpora to strengthen broader problem-solving capabilities on complex, long-context tasks.
Chen Chen, Jiapeng Gao, Gongjie Li, Lingkai Kong, Molei Tao
Transit timing variation (TTV) provides rich information about the mass and orbital properties of exoplanets, which are often obtained by solving an inverse problem via Markov Chain Monte Carlo (MCMC). In this paper, we design a new data-driven approach, which potentially can be applied to problems that are hard to traditional MCMC methods, such as the case with only one planet transiting. Specifically, we use a deep learning approach to predict the parameters of non-transit companion for the single transit system with transit information (i.e., TTV, and Transit Duration Variation (TDV)) as input. This approach is constructed using the Transformer-based architecture that can extract long-range interactions from TTV sequential data. We focus on two-planet systems based on Kepler-88 as a starting point, with the inner planet fixed to the properties of Kepler-88 and the mass and orbital elements of the outer planet varied. Applying to Kepler-88, we obtained an overall fractional error of $\lesssim$10\% on semi-major axis and eccentricity, and $\sim 10-30\%$ on planet mass (assuming that the stellar and transiting planet parameters are known), based on a training set of a million simulations. When applied to the test set, our model achieves low fractional errors of less than a few percent for systems with TTV semi-amplitudes ranging from $\sim 0.3$ s to $\sim 5$ min. Extending our approach to a broader range of planetary architectures, including systems with substantially weaker or stronger TTV signals and those containing additional planets or stellar companions, remains an important direction for future work.