arXiv subjects

Zezheng Song

Publications and source records attributed to Zezheng Song.

6 recordsLinked to original sources

arXiv2026

MathAdv: What Theorem Provers Know, Reason, Formalize, and Generalize

Formal theorem proving enables machine-verifiable evaluation of mathematical reasoning, yet existing benchmarks often emphasize aggregate proof accuracy, concentrate on a narrow range of mathematics, and provide limited evidence of robustness to equivalent reformulations. We introduce MathAdv, a diagnostic benchmark spanning 13 domains across undergraduate- and graduate-level mathematics. Alongside Lean 4 theorem proving, MathAdv provides up to three auxiliary tasks: multiple-choice questions that probe mathematical knowledge, fill-in-the-blank problems that isolate informal reasoning, and expert-crafted transformations that test robustness to problem presentation. Our evaluation of contemporary theorem provers yields four findings: formalization remains a major bottleneck; performance varies substantially across mathematical domains; natural-language guidance helps general-purpose LLMs but can hinder proof-specialized models; and mathematically equivalent reformulations expose substantial robustness limitations. Together, these results show how component-wise evaluation can reveal model capabilities and failure modes that aggregate theorem-proving accuracy obscures. The dataset and evaluation scripts are available at https://github.com/margotyjx/MathAdv.git.

cs.CL↗

arXiv2026

A Finite Expression Method for Solving High-Dimensional Committor Problems

Transition path theory (TPT) is a mathematical framework for quantifying rare transition events between a pair of selected metastable states $A$ and $B$. Central to TPT is the committor function, which describes the probability to hit the metastable state $B$ prior to $A$ from any given starting point of the phase space. Once the committor is computed, the transition channels and the transition rate can be readily found. The committor is the solution to the backward Kolmogorov equation with appropriate boundary conditions. However, solving it is a challenging task in high dimensions due to the need to mesh a whole region of the ambient space. In this work, we explore the finite expression method (FEX, Liang and Yang (2022)) as a tool for computing the committor. FEX approximates the committor by an algebraic expression involving a fixed finite number of nonlinear functions and binary arithmetic operations. The optimal nonlinear functions, the binary operations, and the numerical coefficients in the expression template are found via reinforcement learning. The FEX-based committor solver is tested on several high-dimensional benchmark problems. It gives comparable or better results than neural network-based solvers. Most importantly, FEX is capable of correctly identifying the algebraic structure of the solution which allows one to reduce the committor problem to a low-dimensional one and find the committor with any desired accuracy.

math.NA↗

arXiv2026

A Fast Algorithm for the Finite Expression Method in Learning Dynamics on Complex Networks

Complex network data is prevalent in various real-world domains, including physical, technological, and biological systems. Despite this prevalence, predicting trends and understanding behavioral patterns in complex systems remain challenging due to poorly understood underlying mechanisms. While data-driven methods have advanced in uncovering governing equations from time series data, efforts to extract physical laws from network data are limited and often struggle with incomplete or noisy data. Additionally, they suffer from computational costs on network data, making it difficult to scale to real-world networks. To address these challenges, we introduce a novel approach called the Finite Expression Method (FEX) and its fast algorithm for learning dynamics on complex networks. FEX represents dynamics on complex networks using binary trees composed of finite mathematical operators. The nodes within these trees are trained through a combinatorial optimization process guided by reinforcement learning techniques. This unique configuration allows FEX to capture complex dynamics with minimal prior knowledge of the system and a small dictionary of mathematical operators. We also integrate a fast, stochastic algorithm into FEX, reducing the computational complexity from $O(N^2)$ to $O(N)$. Our extensive numerical experiments demonstrate that FEX excels in accurately identifying dynamics across diverse network topologies and dynamic behaviors.

cs.SC↗

arXiv2025

Omni-Mol: Multitask Molecular Model for Any-to-any Modalities

In the molecular domain, numerous studies have explored the use of multimodal large language models (LLMs) to construct a general-purpose, multi-task molecular model. However, these efforts are still far from achieving a truly universal molecular model. We identify three key challenges in this endeavor: (1) Existing molecular task datasets are typically small in scale and lack comprehensive domain coverage. (2) Tasks from different molecular subfields are difficult to effectively learn jointly through LLMs due to significant distributional shifts and competition among tasks, which introduces instability in the learning process. (3) Both inter-task and intra-task molecular representations demand different intrinsic dimensions in the language space, making it challenging to balance between redundancy and insufficiency in language model representations. To address these challenges, we innovatively categorize existing small-molecule tasks into four types: Mol2Mol, Mol2Text, Mol2Num, and Text2Mol. We then collect a dataset encompassing over 16 tasks with more than 1.4 million samples, making it the largest molecular instruction-tuning dataset to date. Leveraging the extensive pretraining of LLMs on existing chemical literature, we propose a novel multimodal LLM framework, named Omni-Mol, which unifies all small-molecule tasks and supports both molecular generation and understanding. The core of Omni-Mol is our proposed MoGE, which dynamically adapts to the intrinsic rank of different tasks. This mixture-of-experts architecture enhances the model's ability to handle diverse tasks and modalities effectively. Our model achieves unified instruction tuning across 16 tasks and attains state-of-the-art performance on 13 of them. Extensive experiments further demonstrate the scalability and versatility of Omni-Mol.

cs.LG↗

arXiv2024

FMint: Bridging Human Designed and Data Pretrained Models for Differential Equation Foundation Model

The fast simulation of dynamical systems is a key challenge in many scientific and engineering applications, such as weather forecasting, disease control, and drug discovery. With the recent success of deep learning, there is increasing interest in using neural networks to solve differential equations in a data-driven manner. However, existing methods are either limited to specific types of differential equations or require large amounts of data for training. This restricts their practicality in many real-world applications, where data is often scarce or expensive to obtain. To address this, we propose a novel multi-modal foundation model, named \textbf{FMint} (\textbf{F}oundation \textbf{M}odel based on \textbf{In}i\textbf{t}ialization), to bridge the gap between human-designed and data-driven models for the fast simulation of dynamical systems. Built on a decoder-only transformer architecture with in-context learning, FMint utilizes both numerical and textual data to learn a universal error correction scheme for dynamical systems, using prompted sequences of coarse solutions from traditional solvers. The model is pre-trained on a corpus of 40K ODEs, and we perform extensive experiments on challenging ODEs that exhibit chaotic behavior and of high dimensionality. Our results demonstrate the effectiveness of the proposed model in terms of both accuracy and efficiency compared to classical numerical solvers, highlighting FMint's potential as a general-purpose solver for dynamical systems. Our approach achieves an accuracy improvement of 1 to 2 orders of magnitude over state-of-the-art dynamical system simulators, and delivers a 5X speedup compared to traditional numerical algorithms. The code for FMint is available at \url{https://github.com/margotyjx/FMint}.

cs.LG↗

arXiv2024

Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey

This paper explores the recent advancements in enhancing Computational Fluid Dynamics (CFD) tasks through Machine Learning (ML) techniques. We begin by introducing fundamental concepts, traditional methods, and benchmark datasets, then examine the various roles ML plays in improving CFD. The literature systematically reviews papers in recent five years and introduces a novel classification for forward modeling: Data-driven Surrogates, Physics-Informed Surrogates, and ML-assisted Numerical Solutions. Furthermore, we also review the latest ML methods in inverse design and control, offering a novel classification and providing an in-depth discussion. Then we highlight real-world applications of ML for CFD in critical scientific and engineering disciplines, including aerodynamics, combustion, atmosphere & ocean science, biology fluid, plasma, symbolic regression, and reduced order modeling. Besides, we identify key challenges and advocate for future research directions to address these challenges, such as multi-scale representation, physical knowledge encoding, scientific foundation model and automatic scientific discovery. This review serves as a guide for the rapidly expanding ML for CFD community, aiming to inspire insights for future advancements. We draw the conclusion that ML is poised to significantly transform CFD research by enhancing simulation accuracy, reducing computational time, and enabling more complex analyses of fluid dynamics. The paper resources can be viewed at https://github.com/WillDreamer/Awesome-AI4CFD.

cs.LG↗