Most attempts to use deep networks for mathematics

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whether an expression is a valid solution of a given differential equation. Most attempts to use deep networks for mathematics have focused on arithmetic over integers (sometimes over polynomials with integer coefficients). For instance, Kaiser & Sutskever (2015) proposed the Neural-GPU architecture, and train networks to perform additions and multiplications of numbers given in their binary representations. They show that a model trained on numbers with up-to 20 bits can be applied to much larger numbers at test time, while preserving a perfect accuracy. Freivalds & Liepins (2017) proposed an improved version of the Neural-GPU by using hard non-linear activation functions, and a diagonal gating mechanism. Saxton et al. (2019) use LSTMs (Hochreiter & Schmidhuber, 1997) and transformers on a wide range of problems, from arithmetic to simplification of formal expressions. However, they only consider polynomial functions, and the task of differentiation, which is significantly easier than integration. Trask et al. (2018) propose the Neural arithmetic logic units, a new module designed to learn systematic numerical computation, and that can be used within any neural network. Like Kaiser & Sutskever (2015), they show that at inference their model can extrapolate on numbers orders of magnitude larger than the ones seen during training. 6 C ONCLUSION In this paper, we show that standard seq2seq models can be applied to difficult tasks like function integration, or solving differential equations. We propose an approach to generate arbitrarily large datasets of equations, with their associated solutions. We show that a simple transformer model trained on these datasets can perform extremely well both at computing function integrals, and solving differential equations, outperforming state-of-the-art mathematical frameworks like Matlab or Mathematica that rely on a large number of algorithms and heuristics, and a complex implementation (Risch, 1970). Results also show that the model is able to write identical expressions in very different ways. These results are surprising given the difficulty of neural models to perform simpler tasks like integer addition or multiplication. However, proposed hypotheses are sometimes incorrect, and considering multiple beam hypotheses is often necessary to obtain a valid solution. The validity of a solution itself is not provided by the model, but by an external symbolic framework (Meurer et al., 2017). These results suggest that in the future, standard mathematical frameworks may benefit from integrating neural components in their solvers. 12
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R EFERENCES Miltiadis Allamanis, Pankajan Chanthirasegaran, Pushmeet Kohli, and Charles Sutton. Learning con- tinuous semantic representations of symbolic expressions. In Proceedings of the 34th International Conference on Machine Learning - Volume 70 , ICML’17, pp. 80–88., 2017.
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