Lectures on AI for Mathematics

Xiaoyang Chen; Xiaoyang Chen

cs.AIApr 13, 2026

Lectures on AI for Mathematics

Authors: Xiaoyang Chen, Xiaoyang Chen

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 2: Schematic diagram of Principal Component Analysis

Figure 7: Schematic diagram of agent-environment interaction

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

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 $\mathbf{u},\mathbf{v},\mathbf{w}$ )	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

L(\theta)=-\sum_{(s,a)\in\mathcal{D}}\log\pi_{\theta}(a|s)

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

Panel of Figure 14 — Figure 14: Schematic of the 2D finite element method: Left figure shows triangular element mesh, node $i$ connected to several elements; Right figure illustrates the corresponding linear basis function $\phi_{i}(x,y)$ as a “tent function,” taking value $1$ at node $i$ , $0$ on the support boundary, and varying linearly on each adjacent triangular element.

(A_{K})_{ij}=\int_{K}\nabla\phi_{i}\cdot\nabla\phi_{j}\,d\Omega

Algorithm 3 Finite Element Method for Solving 2D Poisson Equation (Triangular Linear Elements)

Panel of Figure 15 — Figure 15: Schematic of 1D finite volume method: Each control volume is centered at node $x_{i}$ , $u_{i}$ is the solution value at node $x_{i}$ , $F_{i\pm 1/2}$ are the fluxes at interfaces $x_{i\pm 1/2}$ , flux exchange occurs at interfaces $x_{i-1/2}$ and $x_{i+1/2}$

a_{\text{conv}}^{\text{left}}=\max(a,0)\quad\text{(upwind scheme)}

Algorithm 4 Finite Volume Method for Solving 1D Steady Convection-Diffusion Equation (Upwind Scheme)

x_{j}=\cos\left(\frac{\pi j}{N}\right),\quad j=0,1,\dots,N

Algorithm 5 Collocation Spectral Method for Solving 1D Poisson Equation (Chebyshev Basis Functions)

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

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