In many computational problems in engineering and science, function or model differentiation is essential, but also integration is needed. An important class of computational problems include so-called integro-differential equations which include both integrals and derivatives of a function. In another example, stochastic differential equations can be written in terms of a partial differential equation of a probability density function of the stochastic variable. To learn characteristics of the stochastic variable based on the density function, specific integral transforms, namely moments, of the density function need to be calculated. Recently, the machine learning paradigm of Physics-Informed Neural Networks emerged with increasing popularity as a method to solve differential equations by leveraging automatic differentiation. In this work, we propose to augment the paradigm of Physics-Informed Neural Networks with automatic integration in order to compute complex integral transforms on trained solutions, and to solve integro-differential equations where integrals are computed on-the-fly during training. Furthermore, we showcase the techniques in various application settings, numerically simulating quantum computer-based neural networks as well as classical neural networks.
翻译:在许多工程和科学的计算问题中,功能或模型差异性是必需的,但也需要整合。 重要的计算问题包括所谓的 Integro - 差异方程式,其中包括函数的构件和衍生物。 在另一个例子中, 随机差异方程式可以用随机变量的概率密度函数的局部差异方程式写成。 要根据密度函数来了解随机变量的特性, 需要计算密度函数的具体整体变换, 即时。 最近, 物理化神经网络的机器学习模式出现了, 越来越受欢迎, 作为一种方法, 利用自动区分法来解决差异方程式。 在这项工作中, 我们提议增加物理化神经网络的范式, 并自动整合, 以便计算经过训练的解决方案的复杂整体变换, 并解决在训练期间在天上计算集成组件的杂交式变异方方程式。 此外, 我们在各个应用环境中展示各种技术, 数字模拟以计算机为基础的神经网络以及古典神经网络。