使用深度神经网络计算反导数
Computing Anti-Derivatives using Deep Neural Networks
摘要：
本文提出了一种新的算法，使用深度神经网络架构获得函数的封闭形式反导数。在过去，数学家已经开发了几种数值技术来近似定积分的值，但原语或不定积分往往是非初等的。当被积函数中有多个参数且所得积分是这些参数的函数时，反导数是必不可少的。对于任何给定的函数，没有理论方法可以做到这一点。一些现有的方法来解决这个问题主要是基于曲线拟合或被积函数的无穷级数近似，然后在理论上积分。曲线拟合近似对于高度非线性函数是不准确的，并且对于每个问题需要不同的方法。另一方面，无穷级数方法不给出封闭形式的解，并且它们的截断形式通常是不准确的。我们声称，使用一个单一的方法为所有的积分，我们的算法可以近似反导数的任何所需的精度。我们用这种算法求出了几个函数的反导数，包括非初等积分和振荡积分。本文还展示了我们的方法在求椭圆积分、Fermi-Dirac积分和累积分布函数的封闭表达式以及减少微分方程Galerkin方法计算时间方面的应用。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
