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The advance of neural ordinary differential ordinary differential equations

Applied and Computational Engineering, 2023
Differential methods are widely used to describe complex continuous processes. The main idea of ordinary differential equations is to treat a specific type of neural network as a discrete equation. Therefore, the differential equation solver can be used to optimize the solution process of the neural network.
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On some nonlinear ordinary differential equations with advanced arguments

Nonlinear Analysis: Theory, Methods & Applications, 2003
The authors consider the following nonlinear differential equation with advanced argument \[ y'(t)= [y(\beta t)]^{1/\beta},\tag{1} \] with \(t\geq 0\) and \(\beta> 1\). By making use of the technique of lower and upper solutions, they classify the solutions of (1) (those that satisfy the initial condition \(y(0)= y_0\) and a certain growth condition ...
Antoni Augustynowicz, Henryk Leszczyński, Wolfgang Walter +2 more
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Oscillations of differential equations generated by advanced arguments

Funkcialaj Ekvacioj, 1987
A nontrivial solution of a differential equation with advanced arguments \[ (1)\quad (-1)^ zx^{(n)}(t)=f(t,x(g_ 1(t)),...,x(g_ m(t)))n\geq 2,\quad z=1,2 \] is called oscillatory if it has an infinite number of zeros \(\to \infty\). Such oscillatory solutions are here studied.
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Advances in Partial Differential Equations and Control