A Neural Network Solves, Explains, and Generates University Math Problems by Program Synthesis and Few-Shot Learning at Human Level (2112.15594v4)

Abstract: We demonstrate that a neural network pre-trained on text and fine-tuned on code solves mathematics course problems, explains solutions, and generates new questions at a human level. We automatically synthesize programs using few-shot learning and OpenAI's Codex transformer and execute them to solve course problems at 81% automatic accuracy. We curate a new dataset of questions from MIT's largest mathematics courses (Single Variable and Multivariable Calculus, Differential Equations, Introduction to Probability and Statistics, Linear Algebra, and Mathematics for Computer Science) and Columbia University's Computational Linear Algebra. We solve questions from a MATH dataset (on Prealgebra, Algebra, Counting and Probability, Intermediate Algebra, Number Theory, and Precalculus), the latest benchmark of advanced mathematics problems designed to assess mathematical reasoning. We randomly sample questions and generate solutions with multiple modalities, including numbers, equations, and plots. The latest GPT-3 LLM pre-trained on text automatically solves only 18.8% of these university questions using zero-shot learning and 30.8% using few-shot learning and the most recent chain of thought prompting. In contrast, program synthesis with few-shot learning using Codex fine-tuned on code generates programs that automatically solve 81% of these questions. Our approach improves the previous state-of-the-art automatic solution accuracy on the benchmark topics from 8.8% to 81.1%. We perform a survey to evaluate the quality and difficulty of generated questions. This work is the first to automatically solve university-level mathematics course questions at a human level and the first work to explain and generate university-level mathematics course questions at scale, a milestone for higher education.

• The paper demonstrates that leveraging program synthesis and few-shot learning enables solving university math problems with 81% accuracy, marking a significant leap over prior methods.
• The approach capitalizes on pre-training across text and code to generate executable programs that facilitate multi-step reasoning in advanced mathematical tasks.
• The network's capability to generate clear explanations and novel problems opens avenues for automated grading, curriculum design, and personalized STEM education.

A Neural Network Solves University Math Problems Using Program Synthesis and Few-Shot Learning

The paper presents a significant advancement in the application of neural networks to solving mathematics problems at a university level by leveraging program synthesis and few-shot learning techniques. The neural network in question, the Codex model developed by OpenAI, is able to solve, explain, and generate questions from various mathematics courses, a feat that underscores both the power and versatility of modern AI when fine-tuned appropriately.

The approach is underpinned by three main innovations: (1) the utilization of neural networks pre-trained on text and further refined on code, (2) the application of few-shot learning to automate the synthesis of programs that solve tasks, and (3) the establishment of a workflow pipeline capable of solving problems, elucidating solutions, and generating novel questions. This paper highlights their methodology as the first AI system capable of addressing university-level mathematics problems across a wide array of subfields at human-level accuracy.

A standout result from the paper is the demonstrated ability of Codex to solve math problems with 81% automatic accuracy, measured against a benchmark consisting of advanced mathematics topics—effectively increasing solution accuracy on this benchmark from 8.8% to 81.1%. These results are particularly notable given that the previous state-of-the-art methods relied solely on text-based pre-training and performed week on mathematical tasks, with GPT-3 achieving only 30.8% accuracy.

The model's success can be attributed to the inherent strength of synthesizing executable programs, which serve as a clear computational representation for solving tasks. Programs also inherently support multi-step reasoning, a computational complexity that chain-of-thought prompting attempts to induce in text-only models. By integrating a substantial corpus of existing code bases during the training process, Codex benefits from the broader utility and expressiveness offered by formal programming languages.

The work has profound implications for the future role of AI in education, particularly in STEM disciplines. Automatic solution generation and question formulation suggest new possibilities for curriculum design, automatic grading systems, and tools for personalized student feedback. Moreover, it raises important considerations regarding educational integrity and the evolving role of educators in environments where AI is increasingly utilized for problem-solving.

The neural network's programmability, which enables modularity, abstraction, and logical illustration through comments and variable names, positions it as a versatile tool beyond math problem-solving. The potential for application expansion to other structured problem domains—mechanics, physics, and potentially even interdisciplinary problems that combine these topics—is substantial.

However, the approach does have limitations, such as its dependency on text-based input and inability to process inputs in purely symbolic or visual forms, such as images or handwritten equations. Additionally, it struggles with proofs, requiring further exploration into formal verification techniques to bridge this gap. Computational scalability is another frontier, wherein future iterations might benefit from improved architectural efficiency or distributed computing approaches to address increasingly complex problem domains.

In summary, the paper represents a notable step forward in the application of AI to mathematics by showcasing a neural network’s ability to solve, explain, and generate university-level math problems with exceptional accuracy. Its implications for education and future AI development are both promising and challenging, framing the discourse for future exploration into AI's role in human learning and knowledge acquisition.