Semi-empirical Neural Network Modeling and Digital Twins Development

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SEMI-EMPIRICAL
NEURAL NETWORK
MODELING AND
DIGITAL TWINS
DEVELOPMENT
Copyright
About the authors
Preface
Acknowledgments
Introduction
	References
1
Examples of problem statements and functionals
	Problems for ordinary differential equations
		A stiff differential equation
		The problem of a chemical reactor
		The problem of a porous catalyst
		Differential-algebraic problem
	Problems for partial differential equations for domains with fixed boundaries
		The Laplace equation on the plane and in space
		The Poisson problem
		The Schrödinger equation with a piecewise potential (quantum dot)
		The nonlinear Schrödinger equation
		Heat transfer in the vessel-tissue system
	Problems for partial differential equations in the case of the domain with variable borders
		Stefan problem
			Problem formulation
		The problem of the alternating pressure calibrator
			Problem statement
	Inverse and other ill-posed problems
		The inverse problem of migration flow modeling
		The problem of the recovery of solutions on the measurements for the Laplace equation
		The problem for the equation of thermal conductivity with time reversal
		The problem of determining the boundary condition
		The problem of continuation of the temperature field according to the measurement data
		Construction of a neural network model of a temperature field according to experimental data in the case of an interval sp ...
		The problem of air pollution in the tunnel
			The conclusion
	References
	Further reading
2
The choice of the functional basis (set of bases)
	Multilayer perceptron
		Structure and activation functions of multilayer perceptron
		The determination of the initial values of the weights of the perceptron
	Networks with radial basis functions-RBF
		The architecture of RBF networks
		Radial basis functions
		Asymmetric RBF-networks
	Multilayer perceptron and RBF-networks with time delays
	References
3
Methods for the selection of parameters and structure of the neural network model
	Structural algorithms
		Methods for specific tasks
	Methods of global non-linear optimization
	Methods in the generalized definition
	Methods of refinement of models of objects described by differential equations
	References
	Further reading
4
Results of computational experiments
	Solving problems for ordinary differential equations
		Stiff form of differential equation
		Chemical reactor problem
		The problem of a porous catalyst
		Differential-algebraic problem
	Solving problems for partial differential equations in domains with constant boundaries
		Solution of the Dirichlet problem for the Laplace equation in the unit circle
		Solving boundary value problems for the Laplace equation in the unit square
		The Laplace equation in the L-region
		The Poisson problem
		Schrödinger equation with a piecewise potential (quantum dot)
		Nonlinear Schrödinger equation
		Heat transfer in the tissue-vessels system
	Solving problems for partial differential equations for domains with variable boundaries
		Stefan problem
		The problem of the variable pressure calibrator
	Solving inverse and other ill-posed problems
		Comparison of neural network and classical approaches to the problem of identification of migration processes
		The problem of the recovery solutions of the Laplace equation on the measurements
		Problem for heat conduction equation with time reversal
		The problem of determining the boundary conditions
		The problem of continuing the temperature field according to measurement data
		Construction of a neural network model of a temperature field in the case of an interval specified thermal conductivity co ...
		The problem of air pollution in a tunnel
	References
5
Methods for constructing multilayer semi-empirical models
	General description of methods
		Explicit methods
		Implicit methods
		Partial differential equations
	Application of methods for constructing approximate analytical solutions for ordinary differential equations
		Comparison of methods on the example of elementary functions
			Results of computational experiments 1: The exponential function
				Error analysis
				Comparison with Maclaurin series with the same number of operations
			Results of computational experiments 2: The cosine function
				Error analysis
				Comparison with the Maclaurin series
				Search of period
		Stiff differential equation
		Mathieu equation
		Nonlinear pendulum equation
			Results of computational experiments for the segment [0;1]
			Results of computational experiments for the segment [0;4]
		The problem of modeling processes in the porous catalyst granule
		Multilayer methods for a model equation with delay
		Application of approximate multilayer methods for solving differential equations in the problem of stabilizing an inverted ...
	Application of multilayer methods for partial differential equations
		Heat equation
		Comparison of multilayer methods for solving the Cauchy problem for the wave equation
	Problems with real measurements
		The problem of sagging hemp rope
		Simulation of the load deflection of the real membrane
		Semi-empirical models of nonlinear bending of a cantilever beam
	References
Index
	A
	B
	C
	D
	E
	F
	G
	H
	I
	J
	K
	L
	M
	N
	O
	P
	Q
	R
	S
	T
	V
	W
	Z
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