Deep Learning Discrete Calculus (DLDC): a family of discrete numerical methods by universal approximation for STEM education to frontier research

The article proposes formulating and codifying a set of applied numerical methods, coined as D eep L earning D iscrete C alculus (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...

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Vydáno v:Computational mechanics Ročník 72; číslo 2; s. 311 - 331
Hlavní autoři: Saha, Sourav, Park, Chanwook, Knapik, Stefan, Guo, Jiachen, Huang, Owen, Liu, Wing Kam
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2023
Springer
Springer Nature B.V
Témata:
Algorithms
Applications of mathematics
Calculus
Classical and Continuum Physics
Codification
Computational Science and Engineering
Data mining
Deep learning
Differential equations
Engineering
Equipment and supplies
Finite difference method
Finite element method
Integral equations
Machine learning
Mathematical analysis
Methods
Numerical analysis
Numerical methods
Original Paper
Teaching
Technical education
Theoretical and Applied Mechanics
Convolution
Numerical methods
Kernel learning
Partial differential equation
Discrete calculus
ISSN:0178-7675, 1432-0924
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Shrnutí:The article proposes formulating and codifying a set of applied numerical methods, coined as D eep L earning D iscrete C alculus (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 on numerical analysis to formulate a general class of numerical methods 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 into 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 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 by a mathematical formulation of the numerical methods, analogous DLDC formulation, and suitable examples.
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ISSN:0178-7675
1432-0924
DOI:10.1007/s00466-023-02292-0