Machine Learning for Mathematics
Apply machine learning techniques to mathematical objects such as polynomials and polytopes.
- Suitable for
- Master's thesis
PhD thesis - Supervision
- Zafeirakis Zafeirakopoulos
We explore how machine learning methods can be applied to mathematical objects (polynomials, polytopes, integer programs) to improve algorithmic performance or guide the discovery of new mathematical structure.
Directions
Symbolic computation via transformers. Lample & Charton (2020) showed that sequence-to-sequence models trained on large datasets of symbolic expressions can solve integration problems and first-order ODEs at competitive speed. This raises the question: which algebraic operations admit efficient neural approximations, and how do they compare to exact methods?
Learning heuristics for combinatorial algorithms. Branch-and-bound solvers, Gröbner basis computations, and root isolation all involve choices (branching, term ordering, subdivision strategy) that are NP-hard to optimize globally but may admit learned heuristics.
Data-driven conjecture discovery. Recent work in algebraic combinatorics uses ML to identify patterns in tabulated data (e.g., Ehrhart polynomials, generating function coefficients) that lead to new conjectures.
Starting point
Replicate the Lample–Charton integration experiment on a small dataset of rational function integrals, then explore whether a similar approach works for polynomial GCD or root isolation sub-problems.
Milestones
| ID | Title |
|---|---|
| M1 | Replicate Lample–Charton on toy dataset |
| M2 | Formulate algebraic task as seq2seq |
| M3 | Benchmark suite + analysis |
| M4 | Final report & release |
Tasks
| ID | Title | Status |
|---|---|---|
| T1 | Replicate Lample–Charton experiment | todo |
| T2 | Design algebraic ML task | todo |
| T3 | Train, evaluate, compare to exact solver | todo |
| T4 | Ablations + write-up | todo |
Deliverables
- Trained model + evaluation code
- Comparison vs. exact symbolic solver
- Report with analysis
References
-
G. Lample and F. Charton. Deep Learning For Symbolic Mathematics. ICLR 2020, OpenReview.net. openreview.net/forum?id=S1eZYeHFDS
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A. Davies, P. Veličković, L. Buesing, S. Blackwell, D. Zheng, N. Tomašev, R. Tanburn, et al. Advancing mathematics by guiding human intuition with AI. Nature, 600:70–74, 2021. DOI 10.1038/s41586-021-04086-x