DeepMath: Pioneering the Convergence of AI and Mathematics Through Technical Innovation

Team Positioning: Pioneers in AI-Mathematics Interdisciplinary Research

Amid the deep integration of artificial intelligence and mathematics, the DeepMath team from Tongji University’s School of Mathematical Sciences has emerged as a benchmark in this field through forward-looking strategic布局 (strategic planning). As one of China’s earliest research forces systematically exploring “AI for Math,” the team takes “mathematical theory innovation driving AI tool evolution” as its core mission, constructing a complete technology stack from foundational theoretical research to engineering implementation. Their research scope covers core scenarios such as mathematical conjecture generation, automated theorem proving, and counterexample construction, forming a dual-engine technical system combining “mathematical large models + intelligent agents.”

Through deep collaboration with organizations like Swarm Intelligence Club, Tsinghua University, and Nanyang Technological University, the team has established an interdisciplinary communication platform. Their regular “AI and Mathematics” themed reading sessions have become a normalized mechanism, organizing over 20 specialized discussions that attracted more than 500 scholars worldwide. This open collaboration model not only accelerated knowledge flow but also produced breakthroughs like the Neural Jacobi Conjecture (NJC), injecting new momentum into traditional mathematical research.

Technical Breakthroughs: From Theory Innovation to Tool Implementation

DeepMath Mathematical Large Model: A Provable Theoretical Tool

The team’s self-developed mathematical large model breaks through traditional AI’s statistical learning paradigm by introducing formal verification mechanisms to ensure interpretable reasoning processes. The model employs a hybrid architecture design:

• Symbolic Reasoning Layer: Constructs a mathematical symbol system based on first-order logic to support rigorous theorem derivation
• Neural Network Layer: Uses Transformer architecture to process unstructured mathematical texts
• Verification Engine: Integrates automated theorem provers for formal validation of generated results

This design enables the model to handle both natural language-described mathematical problems and output verifiable proof processes. In tasks like polynomial equation solving and group theory proofs, the model demonstrates efficiency advantages over traditional symbolic computation systems.

Moonshine Intelligent Agent: A Revolutionary Tool for Conjecture Generation

As the team’s core achievement, the Moonshine intelligent agent redefines mathematical conjecture research paradigms. Its technical architecture comprises three modules:

• Knowledge Graph Construction: Automatically parses concepts, theorems, and relationships from mathematical literature
• Conjecture Generation Engine: Uses Monte Carlo Tree Search to explore potential connections between concepts
• Argument Verification System: Calls the mathematical large model for preliminary validation to filter invalid conjectures

In Jacobi conjecture research, Moonshine creatively migrated the classic question “whether local non-degeneracy implies global injectivity” to single-hidden-layer Affine-ridge Sigmoid networks, proposing the Neural Jacobi Conjecture (NJC). This conjecture has been independently verified by GPT-5.5-pro and DeepSeek-V4-pro for the N=n+1 case but remains unsolved for higher dimensions (N≥n+2), pointing directions for future research.

Technical Implementation: Key Algorithms and Engineering Practices

Mathematical Modeling of the Neural Jacobi Conjecture

The team transformed the traditional Jacobi conjecture into a neural network-processable form:

Let f: R^n → R^n be a single-hidden-layer Affine-ridge Sigmoid network:
f(x) = W_2σ(W_1x + b_1) + b_2
where σ is the Sigmoid function, W_1∈R^{h×n}, W_2∈R^{n×h}
NJC Conjecture: If f has a non-singular Jacobian matrix J_f(x_0) at some point x_0, then f is injective in some neighborhood

By constructing this simplified model, the team successfully converted continuous mathematical problems into discrete optimization problems, providing an entry point for AI intervention.

Implementation of Automated Proof System

For the N=n+1 case proof, the team developed a specialized verification process:

1. Problem Encoding: Converts geometric conditions into algebraic constraints
2. Search Space Pruning: Reduces variable count using symmetry properties
3. Parallel Verification: Distributes proof steps across different computing nodes

Implemented on a GPU cluster from a cloud provider, this process reduced single verification time from weeks (traditional methods) to 72 hours. Code example (pseudocode):

def verify_njc_conjecture(n):
    # Initialize proof environment
    proof_env = ProofEnvironment(dimension=n)
    # Generate candidate proof paths
    paths = generate_proof_paths(proof_env)
    # Parallel verification
    with ThreadPoolExecutor() as executor:
        results = list(executor.map(verify_path, paths))
    # Aggregate verification results
    return any(results)  # Returns True if any path verification succeeds

Ecosystem Construction: Synergistic Development of Research-Education-Tools

The team built a tripartite innovation ecosystem:

• Research System: Published over 30 papers in top journals, establishing theoretical frameworks for AI mathematics research
• Educational Practice: Offers interdisciplinary courses in “Artificial Intelligence and Mathematics” to cultivate compound talents
• Tool Development: Open-sourced core Moonshine modules, receiving 2,000+ GitHub stars

This synergistic model produced significant spillover effects: Their monograph “AI for Math: From Pattern Discovery to Theorem Proving” systematically梳理 (organizes) interdisciplinary research methodologies, becoming an important reference in the field. The proposed “mathematical problem formal representation framework” has been adopted as a standard by multiple research institutions.

Future Prospects: Challenges and Opportunities Coexist

Despite achieving phased results, the team faces multiple challenges:

1. High-Dimensional Breakthroughs: Proofs for N≥n+2 require stronger symbolic reasoning capabilities
2. Tool Versatility: Improving model adaptability to algebraic geometry, number theory, and other fields
3. Ethical Risk Prevention: Establishing credibility assessment systems for AI-generated mathematical content

To address these challenges, the team is exploring collaboration with a cloud provider to build a distributed proof network. This network will integrate global computing resources to form ultra-large-scale parallel verification capabilities, offering new possibilities for solving mathematical challenges like Hilbert’s 23rd Problem.

The DeepMath team’s practice demonstrates that the deep integration of AI and mathematics is not merely tool substitution but creates entirely new research paradigms. As breakthroughs occur in key technologies like neurosymbolic systems and large model interpretability, this interdisciplinary revolution is reshaping the future landscape of mathematical research. For researchers, mastering such cross-disciplinary methodologies will become core competence for international competition; for developers, AI tool development guided by mathematical theory will open new spaces for value creation.