Deep Learning for Symbolic Mathematics (1912.01412v1)

Abstract: Neural networks have a reputation for being better at solving statistical or approximate problems than at performing calculations or working with symbolic data. In this paper, we show that they can be surprisingly good at more elaborated tasks in mathematics, such as symbolic integration and solving differential equations. We propose a syntax for representing mathematical problems, and methods for generating large datasets that can be used to train sequence-to-sequence models. We achieve results that outperform commercial Computer Algebra Systems such as Matlab or Mathematica.

• The paper demonstrates that transformer-based seq2seq models can outperform commercial algebra systems in symbolic integration and differential equation solving.
• It introduces a novel mathematical syntax and dataset generation method to train models on a vast array of symbolic problems.
• The research provides insights into model generalization, paving the way for hybrid approaches that integrate symbolic reasoning with deep learning.

Overview of Deep Learning for Symbolic Mathematics

The paper "Deep Learning for Symbolic Mathematics" by Guillaume Lample and François Charton explores the capability of neural networks, typically associated with statistical approximation, to engage effectively in symbolic mathematics tasks such as symbolic integration and solving differential equations. Despite the prevailing belief that neural networks underperform in symbolic computation compared to statistical tasks, this research demonstrates that sequence-to-sequence (seq2seq) models can outperform commercial Computer Algebra Systems like Matlab and Mathematica in these tasks under certain conditions.

Core Contributions

1. Mathematical Syntax and Dataset Generation: A primary contribution is the development of a representation system for mathematical problems conducive to seq2seq models, along with methods for generating extensive datasets to train these models. The paper introduces algorithms to generate these expressions and offers detailed measurement of problem space size using combinatorial methods.
2. Experiments and Model Performance: Through comprehensive experiments, the paper shows that transformer-based seq2seq models can approximate and solve symbolic mathematics problems with exceptional accuracy on tasks where datasets were generated. The models performed comparably or superior to existing software solutions, suggesting that machine learning approaches can augment traditional symbolic frameworks.
3. Generalization and Cross-Validation: An analysis of generalization across different types of datasets reveals significant insights into how models trained on one type of problem space might perform on another. This involves integrating data from forward generation (FWD), backward generation (BWD), and integration by parts (IBP), with each approach having its unique characteristics in generating expressions.

Implications for AI and Future Prospects

1. Theoretical and Practical Implications: This work challenges the prevailing notion of neural networks' limitation to statistical learning without capability in symbolic reasoning. The insights derived here suggest that neural networks could integrate into symbolic computational tools, potentially simplifying algorithmic complexity inherent in systems like the Risch algorithm for symbolic integration.
2. Future Research Directions: The findings advocate for further exploration into hybrid models that can seamlessly intersperse symbolic reasoning within statistical contexts, bridging the gap in solvability and flexibility of mathematical problems. There is also a call to investigate the efficacy of models on out-of-distribution problems and the development of robust evaluation frameworks that extend beyond model accuracy.

In sum, this paper not only showcases the plausibility of using neural networks for symbolic mathematics tasks but also posits considerable potential augments to traditional CAS frameworks. This represents a step towards the broader applicability of deep learning in AI's understanding and processing of complex mathematical expressions, potentially leading to automated theorem proving and other advanced computational tasks in the future.