cs.AIApr 13, 2026
SaveLectures on AI for Mathematics
Organizations: School of Mathematical Sciences, Tongji University
Abstract
This book provides a comprehensive and accessible introduction to the emerging field of AI for mathematics. It covers the core principles and diverse applications of using artificial intelligence to advance mathematical research. Through clear explanations, the text explores how AI can discover hidden mathematical patterns, assist in proving complicated theorems, and even construct counterexamples to challenge conjectures.
Figures & tables
Figure 1: Support Vector Machine
Figure 2: Schematic diagram of Principal Component Analysis
Figure 3: Neuron Structure
Figure 4: Feedforward Neural Network
Figure 5: Convolutional Layer
Figure 6: Recurrent Neural Network
Figure 7: Schematic diagram of agent-environment interaction
Figure 8: Markov diagram
Figure 9: Generative Adversarial Network
Figure 10: Attention computation process
Figure 11: Transformer network architecture
| Stage | Learning Paradigm | Source of Supervision Signal | Core Objective |
| Pretraining | Self-Supervised Learning | Structure of the data itself | Learn language patterns and world knowledge, establish foundational capabilities |
| Supervised Fine-Tuning | Supervised Learning | Annotations | Teach the model to follow instructions and output formats |
| Reinforcement Learning | Reinforcement Learning | Reward signal | Optimize output quality, align with human preferences and values |
Table 1: Three Reinforcement Learning Paradigms
Figure 12: AlphaEvolve Workflow
| Dimension | AlphaTensor | AlphaEvolve |
| Core Method | Deep Reinforcement Learning (based on AlphaZero/MCTS), customized for a specific problem formalization (game) | Large Language Model-guided program evolution, employing a general problem-solving framework |
| Problem Formalization | Formalizes “finding decomposition” as a single-player game (TensorGame), state is a tensor, action is a rank-one factor | Formalizes “solving the problem” as a code optimization task, state/knowledge is implicit in the program database, action is code modification |
| Search Space | Directly searches the solution space (decomposition factors ) | Searches the meta-space (programs that can generate or find solutions) |
| Core Technology | Specialized neural network (Transformer variant designed for tensors), trained via reinforcement learning | General large language model (e.g., Gemini), its knowledge comes from pre-training, invoked via prompt engineering and in-context learning |
| Advantages | Highly targeted, high search efficiency; good theoretical guarantees; extreme performance within the target domain | Extremely flexible, requires almost no modification of the core framework for different problems; can handle complex, multi-component codebases; can seamlessly incorporate rich natural language domain knowledge; outputs are often more interpretable and deployable |
| Limitations | Requires designing specialized game rules, state representations, and reward functions for each new problem; neural network architecture is tightly coupled with the problem, high migration cost | Heavily reliant on the existence and quality of the automatic evaluation function; evolutionary process can be slow and depends on LLM generation quality; may have high computational resource requirements |
Table 2: Methodological Comparison of AlphaTensor and AlphaEvolve
Algorithm 1 Deep Cross-Entropy Method
Figure 13: Reinforcement Learning Constructing Counterexamples
| Method Name | Core Discretization Idea | Typical Application Fields | Key Advantages | Main Challenges |
| Finite Difference Method (FDM) | Approximate derivatives using difference quotients of function values at grid points | Fluids, heat conduction on regular domains | Simple concept, easy implementation, mature theory | Poor adaptation to complex geometry, curse of dimensionality |
| Finite Element Method (FEM) | Partition domain into elements, approximate solution using local polynomial basis functions, based on variational principles | Structural mechanics, fluids with complex geometry, electromagnetics | Strong geometric adaptability, rigorous theory, widest application | Complex mesh generation, high computational cost in high dimensions |
| Finite Volume Method (FVM) | Divide control volumes, integrate conservation laws, maintain physical flux balance | Computational fluid dynamics, combustion, multiphase flow | Strictly maintains physical conservation, strong robustness | Complex high-order schemes, difficult theoretical analysis |
| Spectral Method | Expand solution using global smooth basis functions (e.g., Fourier series, Chebyshev polynomials) | Turbulence simulation, quantum chemistry, weather forecasting | Exponential convergence accuracy for smooth solutions | Requires smooth solution, poor geometric adaptability, dense matrices |
Table 3: Comparison of Four Mainstream Numerical Methods
Figure 14: Schematic of the 2D finite element method: Left figure shows triangular element mesh, node connected to several elements; Right figure illustrates the corresponding linear basis function as a “tent function,” taking value at node , on the support boundary, and varying linearly on each adjacent triangular element.
Algorithm 3 Finite Element Method for Solving 2D Poisson Equation (Triangular Linear Elements)
Figure 15: Schematic of 1D finite volume method: Each control volume is centered at node , is the solution value at node , are the fluxes at interfaces , flux exchange occurs at interfaces and
Algorithm 4 Finite Volume Method for Solving 1D Steady Convection-Diffusion Equation (Upwind Scheme)
Algorithm 5 Collocation Spectral Method for Solving 1D Poisson Equation (Chebyshev Basis Functions)
Figure 16: PINN Training Process
| Architecture Type | Core Idea | Applicable Scenarios | Advantages | Disadvantages and Challenges |
| MLP | Fully connected, universal function approximator | General, low-dimensional problems, irregular geometry | Solid theoretical guarantee (Universal Approximation Theorem), simple implementation, naturally compatible with automatic differentiation | Suffers from “spectral bias”, low parameter efficiency for high-dimensional problems |
| CNN | Local connectivity, weight sharing, translation invariance | Regular grid data (e.g., image-like physical fields), problems with local correlations | High parameter efficiency, effective at extracting local features, suitable for high-dimensional spatial processing | Complex to handle irregular geometry, requires projecting input onto regular grids |
| RNN/LSTM | Recurrent connections, memory of historical states | Strongly time-dependent problems, sequence prediction | Naturally suited for temporal modeling, can capture long-term dependencies | Training prone to gradient vanishing/explosion, low parallelization |
| Transformer | Self-attention mechanism, global dependency modeling | Complex geometry, long-range interactions, point cloud data | Can model dependencies at arbitrary distances, high flexibility | High computational and memory cost, training requires large amounts of data |
| KAN | Learnable activation functions, based on Kolmogorov-Arnold theorem | Function approximation, scenarios requiring high interpretability | Fewer parameters, potentially better interpretability | Still in exploratory stage, theory and methods are still developing |
Table 4: Comparison of Commonly Used Neural Network Architectures in PINNs