Deep Learning Discrete Calculus (DLDC): A Family of Discrete Numerical Methods by Universal Approximation for STEM Education to Frontier Research

TL;DR

This paper introduces Deep Learning Discrete Calculus (DLDC), a novel framework combining deep learning and numerical methods to improve the interpretation, prediction, and solution of complex systems in STEM education and research.

Contribution

It formulates a new class of numerical methods using deep learning that can handle data uncertainty and enhance solution speed and accuracy for differential and integral equations.

Findings

Developed DLDC methods analogous to finite difference and finite element methods.

Demonstrated DLDC's effectiveness in multiscale mechanics problems.

Proposed integrating DLDC into STEM education for K-12 students.

Abstract

The article proposes formulating and codifying a set of applied numerical methods, coined as Deep Learning Discrete Calculus (DLDC), that uses the knowledge from discrete numerical methods to interpret the deep learning algorithms through the lens of applied mathematics. The DLDC methods aim to leverage the flexibility and ever increasing resources of deep learning and rich literature of numerical analysis to formulate a general class of numerical method that can directly use data with uncertainty to predict the behavior of an unknown system as well as elevate the speed and accuracy of numerical solution of the governing equations for known systems. The article is structured in two major sections. In the first section, the building blocks of the DLDC methods are presented and deep learning structures analogous to traditional numerical methods such as finite difference and finite element methods are constructed with a view to incorporate these techniques in Science, Technology, Engineering, Mathematics (STEM) syllabus for K-12 students. The second section builds upon the building blocks of the previous discussion,and proposes new solution schemes for differential and integral equations pertinent to multiscale mechanics. Each section is accompanied with mathematical formulation of the numerical methods, analogous DLDC formulation, and suitable examples.