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The Use of Deep Learning for Symbolic Integration: A Review of (Lample and Charton, 2019)

Paper · 2019 · Ernest Davis

Summary

Ernest Davis critically reviews the system by Lample and Charton (2019), called LC here. LC frames symbolic indefinite integration, and the solving of first- and second-order ODEs, as a translation task: a seq2seq transformer maps an integrand to its integral. It was trained on 80 million synthetic pairs from three generators (FWD, BWD, IBP). At test time, beam-search candidates are checked by differentiating them and simplifying the result.

On a test set built only from BWD examples, LC solved far more integrals than Mathematica, Maple or Matlab. Davis grants that this is impressive but argues the claim of beating Mathematica needs heavy qualification. The test problems form a narrow niche: very long integrands whose elementary integrals are short. LC was never tested on integrands that need simplification first, on special functions, on symbolic parameters, or on non-elementary integrals, where it always returns some wrong answer. It also transfers poorly between data generators and depends fully on conventional computer algebra. Davis concludes the result shows little about deep learning replacing symbolic systems or about real-world reasoning.

Key points

  • Training corpus (80M pairs): FWD, 20M pairs (an existing integrator solves random functions); BWD, 40M pairs (random functions are differentiated); IBP, 20M pairs (integration by parts).
  • Integration results: On the BWD-only test set, Mathematica solved 84% and LC with beam size 1 solved 98.4%. Test integrands averaged 70 symbols (SD 47).
  • ODE results:
    • First-order: LC 81.2% vs Mathematica 77.2% at beam 1, and 97% at beam 50.
    • Second-order: LC 40.8% vs Mathematica 61.6% at beam 1, and 81.0% at beam 50.
  • Weak cross-generator transfer:
    • Trained on BWD, tested on FWD: 18.9% (beam 1) and 27.5% (beam 50).
    • Trained on FWD, tested on BWD: 10.9% and 17.2%.
  • Noted limitations:
    • Only one variable x, with constants limited to integers −5 to 5.
    • No simplification step, confirmed by François Charton.
    • Possible dependence on the fixed term order produced by the differentiator.
    • No ability to say that no solution exists.

Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.

 Institutions

🎓New York UniversityNew York · United States233 works · 104,003 citations

 Fields, topics and concepts

📚 Shelves 4The field of science this work belongs to (OpenAlex fields).Computer Science · 19,503 worksMathematics · 233 worksComputational Theory and Mathematics · 496 worksModeling and Simulation · 12 works

Cite

Cite as Ernest Davis. 2019. The Use of Deep Learning for Symbolic Integration: A Review of (Lample and Charton, 2019). arXiv:1912.05752.

The paper itself, not this page. BibTeX from the library's record of the work. DOI 10.48550/arxiv.1912.05752 · arXiv 1912.05752

@misc{davis2019use,
  title = {{The Use of Deep Learning for Symbolic Integration: A Review of (Lample and Charton, 2019)}},
  author = {Ernest Davis},
  year = {2019},
  doi = {10.48550/arxiv.1912.05752},
  eprint = {1912.05752},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.48550/arxiv.1912.05752}
}

References

3 references, 2 of them with a card of ours. ↗ leads to the work, «find» searches for one we have no id for yet.

  1. Lample, Guillaume and François Charton, 2019. NeurIPS-2019. https://arxiv.org/abs/1912.01412 ↗
  2. Richardson, Daniel, 1969. “Some undecidable problems involving elementary functions of a real variable.” The Journal of Symbolic Logic, 33:4, 514-520. find
  3. WolframAlpha, “Integrals that Can and Cannot be Done” — Deep Learning for Symbolic Mathematics https://reference.wolfram.com/language/tutorial/IntegralsThatCanAndCannotBeDone.html WolframAlpha, “Symbolic Differential Equation Solving” https://reference.wolfram.com/language/tutorial/DSolveOverview.html “Deep Learning for Symbolic Mathematic… ↗

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