Computing Anti-Derivatives using Deep Neural Networks

TL;DR

This paper introduces a deep neural network-based algorithm capable of computing closed-form anti-derivatives for a wide range of functions, including non-elementary and oscillatory integrals, with high accuracy.

Contribution

The paper presents a novel, universal neural network approach for obtaining anti-derivatives, overcoming limitations of existing numerical and series-based methods.

Findings

Successfully computed anti-derivatives of non-elementary functions

Achieved closed-form expressions for elliptic and Fermi-Dirac integrals

Reduced computation time in differential equation solutions

Abstract

This paper presents a novel algorithm to obtain the closed-form anti-derivative of a function using Deep Neural Network architecture. In the past, mathematicians have developed several numerical techniques to approximate the values of definite integrals, but primitives or indefinite integrals are often non-elementary. Anti-derivatives are necessarily required when there are several parameters in an integrand and the integral obtained is a function of those parameters. There is no theoretical method that can do this for any given function. Some existing ways to get around this are primarily based on either curve fitting or infinite series approximation of the integrand, which is then integrated theoretically. Curve fitting approximations are inaccurate for highly non-linear functions and require a different approach for every problem. On the other hand, the infinite series approach does not give a closed-form solution, and their truncated forms are often inaccurate. We claim that using a single method for all integrals, our algorithm can approximate anti-derivatives to any required accuracy. We have used this algorithm to obtain the anti-derivatives of several functions, including non-elementary and oscillatory integrals. This paper also shows the applications of our method to get the closed-form expressions of elliptic integrals, Fermi-Dirac integrals, and cumulative distribution functions and decrease the computation time of the Galerkin method for differential equations.