scholarly journals Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control

2021 ◽  
Vol 43 (4) ◽  
pp. A2474-A2501
Author(s):  
Mark Ainsworth ◽  
Justin Dong
Keyword(s):  
Neural Networks ◽  
Error Control ◽  
Variational Equations

A new neural network approach to density calculation on ceramic materials

2021 ◽  
Author(s):  
Vojislav V. Mitic ◽  
Srdjan Ribar ◽  
Branislav M. Randjelovic ◽  
Dejan Aleksic ◽  
Hans Fecht ◽  
...  
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Error Control ◽  
Back Propagation ◽  
Ceramic Materials ◽  
Training Procedure ◽  
Neural Network Approach ◽  
Network Output ◽  
To Come ◽  
Error Generation

The materials’ consolidation, especially ceramics, is very important in advanced research development and industrial technologies. Science of sintering with all incoming novelties is the base of all these processes. A very important question in all of this is how to get the more precise structure parameters within the morphology of different ceramic materials. In that sense, the advanced procedure in collecting precise data in submicro-processes is also in direction of advanced miniaturization. Our research, based on different electrophysical parameters, like relative capacitance, breakdown voltage, and [Formula: see text], has been used in neural networks and graph theory successful applications. We extended furthermore our neural network back propagation (BP) on sintering parameters’ data. Prognosed mapping we can succeed if we use the coefficients, implemented by the training procedure. In this paper, we continue to apply the novelty from the previous research, where the error is calculated as a difference between the designed and actual network output. So, the weight coefficients contribute in error generation. We used the experimental data of sintered materials’ density, measured and calculated in the bulk, and developed possibility to calculate the materials’ density inside of consolidated structures. The BP procedure here is like a tool to come down between the layers, with much more precise materials’ density, in the points on morphology, which are interesting for different microstructure developments and applications. We practically replaced the errors’ network by density values, from ceramic consolidation. Our neural networks’ application novelty is successfully applied within the experimental ceramic material density [Formula: see text] [kg/m3], confirming the direction way to implement this procedure in other density cases. There are many different mathematical tools or tools from the field of artificial intelligence that can be used in such or similar applications. We choose to use artificial neural networks because of their simplicity and their self-improvement process, through BP error control. All of this contributes to the great improvement in the whole research and science of sintering technology, which is important for collecting more efficient and faster results.

On the motion of comet P/d’Arrest

1974 ◽  
Vol 22 ◽  
pp. 145-148
Author(s):  
W. J. Klepczynski
Keyword(s):  
Equations Of Motion ◽  
Variational Equations ◽  
Partial Derivatives

AbstractThe differences between numerically approximated partial derivatives and partial derivatives obtained by integrating the variational equations are computed for Comet P/d’Arrest. The effect of errors in the IAU adopted system of masses, normally used in the integration of the equations of motion of comets of this type, is investigated. It is concluded that the resulting effects are negligible when compared with the observed discrepancies in the motion of this comet.

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