
Deep Learning outperforms Mathematica on symbolic integration and solving ODEs - htfy96
https://arxiv.org/abs/1912.01412
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htfy96
Author's original tweet:
[https://twitter.com/GuillaumeLample/status/12021789560630640...](https://twitter.com/GuillaumeLample/status/1202178956063064064)

> Although neural networks struggle on simple arithmetic tasks such as
> addition and multiplication, we show that transformers perform surprisingly
> well on difficult mathematical problems such as function integration and
> differential equations. > We define a general framework to adapt seq2seq
> models to various mathematical problems, and present different techniques to
> generate arbitrarily large datasets of functions with their integrals, and
> differential equations with their solutions. > On samples of randomly
> generated functions, we show that transformers achieve state-of-the-art
> performance and outperform computer algebra systems such as Mathematica. >
> We show that beam search can generate alternative solutions for a
> differential equation, all equivalent, but written in very different ways.
> The model was never trained to do this, but managed to figure out that
> different expressions correspond to the same mathematical object > We also
> observe that a transformer trained on functions that SymPy can integrate, is
> able at test time to integrate functions that SymPy is not able to
> integrate, i.e. the model was able to generalize beyond the set of functions
> integrable by SymPy. > A purely neural approach is not sufficient, since it
> still requires a symbolic framework to check generated hypotheses. Yet, our
> models perform best on very long inputs, where computer algebra systems
> struggle. Symbolic computation may benefit from hybrid approaches.

