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 multiplicationsof numbers given in their binary representations. They show that a model trained on numbers withup-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-linearactivation functions, and a diagonal gating mechanism.Saxton et al. (2019) use LSTMs (Hochreiter & Schmidhuber, 1997) and transformers on a widerange of problems, from arithmetic to simplification of formal expressions. However, they onlyconsider polynomial functions, and the task of differentiation, which is significantly easier thanintegration. Trask et al. (2018) propose the Neural arithmetic logic units, a new module designed tolearn 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 ofmagnitude larger than the ones seen during training.6CONCLUSIONIn this paper, we show that standard seq2seq models can be applied to difficult tasks like functionintegration, or solving differential equations. We propose an approach to generate arbitrarily largedatasets of equations, with their associated solutions. We show that a simple transformer modeltrained on these datasets can perform extremely well both at computing function integrals, andsolving differential equations, outperforming state-of-the-art mathematical frameworks like Matlab orMathematica 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 differentways.These results are surprising given the difficulty of neural models to perform simpler tasks like integeraddition or multiplication. However, proposed hypotheses are sometimes incorrect, and consideringmultiple beam hypotheses is often necessary to obtain a valid solution. The validity of a solution itselfis not provided by the model, but by an external symbolic framework (Meurer et al., 2017). Theseresults suggest that in the future, standard mathematical frameworks may benefit from integratingneural components in their solvers.12
REFERENCESMiltiadis Allamanis, Pankajan Chanthirasegaran, Pushmeet Kohli, and Charles Sutton. Learning con-tinuous semantic representations of symbolic expressions. InProceedings of the 34th InternationalConference on Machine Learning - Volume 70, ICML’17, pp. 80–88. JMLR.org, 2017.
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