The classic principle of reduction suggests removing unnecessary elements of the model one at a time [2]. The reduction process can be significantly accelerated by group removal of elements. The article substantiates such removal and shows its effectiveness on the example of the identification of the Lorenz ODE.
But this property can not detect unnecessary parameters, because some of the needed
parameters may be close to zero. Next, we apply slight random variations — referred to as
disturbances — in a manner that preserves the system’s continuous dependence on the solution.
The identification algorithm then determines a parameter vector p′p'p′ corresponding to the
disturbed system, which differs from the original parameter vector ppp of the undisturbed system.
For each parameter, we calculate the absolute values of relative deviations (RD) using the
following approach: :
For the relevant parameters, the absolute differences (pi′−pi)(p'_i - p_i)(pi′−pi) approach zero as
the magnitude of the perturbations diminishes, owing to the continuous relationship between the
parameters and the introduced disturbances. A similar behavior is observed for the relative
deviations (RD):
δi = abs(( p'i − pi) / p'i ); i=1,…,m.
(
The classical equations of the Lorentz attractor (
δi → 0; i = 1,…,n; if disturb → 0.
δi → 1; i = n+1,…,m; if disturb ≠ 0
Contrary, for the unnecessary parameters the values of the RD (
Thus, the task of recovering the exact model (6) can be formulated as follows: given a discrete signal y1= x1, it is necessary to compute its first, second, and third derivatives. y1'= y2, y2'= y3, y3', and solve the identification problem (7). (7)
All polynomial coefficients of the problem (7) are 50. But for the exact model (6) only 7 coefficients are required. The remaining coefficients are unnecessary.
The polynomial representation in problem (7) includes 50 coefficients. However, the accurate
model described in (6) requires only 7 of them; the rest are redundant and do not contribute to the
solution. The discrete signal y1=x1 was obtained by applying numerical integration to equations (
In the reduction process, the magnitude of the RD max(δi)-min(δi), and the middle relative error model coefficients were calculated. After 43 reduction steps, there are 7 coefficients of the exact model (6) with an middle relative error 0.0016. In Fig. 1 shows a change in relative error with increasing number of reduction step. 5 10 15 20 25 30 35 40 45 The dependence of the area size of relative deviations on the reduction step is shown in Fig. 2.
The same figure on an enlarged scale Fig 3. is demonstrated. The process of forming a compact area is well visible.
The model expansion approach (induction) was applied to the Lorenz system to evaluate its
effectiveness. Relative deviations (as defined in equation (
The benefits of the induction approach over reduction are evident. Firstly, there is no need to recalculate relative deviations at every stage—initial computation suffices for the entire process. Secondly, the total number of iterations may be reduced.
Thus, using the Lorenz attractor as a test case, the fundamental principles underpinning the reduction method were effectively validated.
Removing model elements one at a time at each reduction step requires too much time to reduce complex models with a large number of elements. The reduction process can be accelerated by group removal of elements. You just need to make sure that the group you are deleting does not have the necessary model elements.
Fig. 2 shows that the size of the relative deviation region (RDR) of elements can change significantly during the reduction process. In the test case of the Lorentz attractor the necessary elements are known, and it is possible to study the change in their RD in the process of reduction. It turned out that the RD of the necessary elements is always less than the middle of the RDR at any of its sizes, The same conclusion is obtained for other test models [5].
This means that you can delete a group of elements with an RD greater than the middle of the RDR at any reduction step except for the last ones, when this criterion does not work. It is only necessary to select the minimum size of the group to be removed in order to successfully complete the reduction process.
Fig. 4 shows the change in the error of the model coefficients when deleting elements with a minimum group size of 4. The lacunaes correspond to three group deletions with group sizes of 13, 4, and 4 elements. The reduction process accelerated by 49%. If you reduce the minimum size of the deleted group to 3, the reduction process is not successful.
In the publication [6], on the basis of the principle of reduction, group removal of elements without any justification is proposed. 5 10 15 20 25 30 35 40 45
The improvement of mathematical models using the principle of reduction is proposed. The improvement consists in the group removal of unnecessary elements of models.
Using the example of a test model, it is shown how to form a group of deleted elements without the necessary model elements.
Declaration on Generative AI During the preparation of this work, the authors utilised ChatGPT and LanguageTool to identify and rectify grammatical, typographical, and spelling errors. Following the use of these tools, the authors conducted a thorough review and made necessary revisions, and accept full responsibility for the final content of this publication.