For certain families of multivariable vector-valued functions to be approximated, the accuracy of approximation schemes made up of linear combinations of computational units containing adjustable parameters is investigated.
The obtained results reveals that the approximation speed of the constructed FNNs depends not only on the number of hidden units used, but also on the smoothness of functions to be approximated.
We present, by making use of multivariate approximation tools, a quantitative upper bound estimation on approximation accuracy of the simultaneous approximation of the FNNs in terms of the modulus of smoothness of the functions to be approximated.
In practice, the Heaviside function H z) needs to be approximated by a smooth function H ε z, which is typically given by H ε z = 1 2 1 + 2 π arctan z ε (23).
For practical and feasible implementation, the Heaviside function H z) has to be approximated by a smooth function H ε (z) given typically by H ε z = 1 2 1 + 2 π arctan z ε (20).
The net current distribution function is found to be approximated well by a Gaussian and the average magnitude of the net currents is defined by higher than linear order even terms in the response function.
On the other hand, the energy window setup resulted in a complicated time resolution function, which had to be approximated by three Gaussians with FWHM 0.28, 0.35, and 0.75 ns, and intensity of 69%, 188 %, and 13%%, respectively.
The dressing functions A g, i have to be approximated, e.g., by discretization of the argument or expansion in an orthogonal set of functions (see Section 4.1).
The lengths and coefficients of these trips can then be calculated from their full trip descriptions, allowing the Green's function given by Eq. (5) to be approximated.
