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Symmetry
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Symmetry in Mathematical Models
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Symmetry in Mathematical Models

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 31 October 2024 | Viewed by 4371

Special Issue Editors

Dr. Dušan Nikezić

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Guest Editor
Vinča Institute of Nuclear Sciences - National Institute of the Republic of Serbia, University of Belgrade, Belgrade, Serbia
Interests: machine learning; deep learning; CNN; LSTM; remote sensing data; mathematical modelling
Prof. Dr. Vesna Borka Jovanović

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Guest Editor
Department of Theoretical Physics and Condensed Matter Physics (020), Vinča Institute of Nuclear Sciences - National Institute of the Republic of Serbia, University of Belgrade, P.O. Box 522, 11001 Belgrade, Serbia
Interests: radioastronomy; mass spectra of elementary particles; numerical experiments; big data; gravitation and the structure of the Universe

Special Issue Information

Dear Colleagues,

Symmetry represents agreement in dimensions due to proportionality and refers to a sense of harmonious proportionality and balance. In mathematics, symmetry has a precise definition, that an object is invariant to any of a variety of transformations, including reflection, rotation, or scaling. In geometry, symmetry is the mirroring (mapping) of figures. Symmetry is the property of a symmetrical figure in relation to a line (axis), point (center) or plane. Integers are said to be symmetric (palindromes) if they are read the same on both the left and right sides. Biosymmetry studies the symmetry of biostructures at the molecular and supramolecular level and allows the determination, in advance, of the possible variants of symmetry in biological objects, strictly describing the external form and internal structure of any organism. Only two main types of symmetry are known: rotational and translational, or there is a modification from the combination of these two basic types of symmetry rotational–translational symmetry.

Therefore, the aim of this Special Issue is to showcase a range of proposed multidisciplinary studies on symmetry. Here, we want to point out that the proper mathematical model can explain it, and this Special Issue will be a collection of a number of such examples.

Dr. Dušan Nikezić
Prof. Dr. Vesna Borka Jovanović
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • artificial intelligence
  • machine learning
  • neural network
  • data analysis
  • big data in astrophysics
  • nonlinear dynamics
  • biomathematics
  • mathematical modelling
  • chaos
  • symmetry

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Published Papers (5 papers)

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Research

25 pages, 577 KiB  
Article
Invariant Sets, Global Dynamics, and the Neimark–Sacker Bifurcation in the Evolutionary Ricker Model
by Rafael Luís and Brian Ryals
Symmetry 2024, 16(9), 1139; https://doi.org/10.3390/sym16091139 - 2 Sep 2024
Viewed by 465
Abstract
In this paper, we study the local, global, and bifurcation properties of a planar nonlinear asymmetric discrete model of Ricker type that is derived from a Darwinian evolution strategy based on evolutionary game theory. We make a change of variables to both reduce [...] Read more.
In this paper, we study the local, global, and bifurcation properties of a planar nonlinear asymmetric discrete model of Ricker type that is derived from a Darwinian evolution strategy based on evolutionary game theory. We make a change of variables to both reduce the number of parameters as well as bring symmetry to the isoclines of the mapping. With this new model, we demonstrate the existence of a forward invariant and globally attracting set where all the dynamics occur. In this set, the model possesses two symmetric fixed points: the origin, which is always a saddle fixed point, and an interior fixed point that may be globally asymptotically stable. Moreover, we observe the presence of a supercritical Neimark–Sacker bifurcation, a phenomenon that is not present in the original non-evolutionary model. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Models)
►▼ Show Figures

Figure 1

Figure 1
<p>The graph of <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>C</mi> </mrow> </semantics></math> separates the plane into two distinct regions. All points in the interior of the shaded region have two distinct preimages. Points in the unshaded region have no preimage.</p> Full article ">Figure 2
<p>Graphs of <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>C</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>L</mi> <msub> <mi>C</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msup> <mi>F</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mi>L</mi> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, and a line <span class="html-italic">L</span> with slope <math display="inline"><semantics> <mrow> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msup> <mi>σ</mi> <mn>2</mn> </msup> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>σ</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mstyle> </mrow> </semantics></math> and its image under <span class="html-italic">F</span> are shown. The image of the line is symmetric about the point on <math display="inline"><semantics> <mrow> <mi>L</mi> <msub> <mi>C</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math>, where each point to the lower right of <math display="inline"><semantics> <mrow> <mi>L</mi> <msub> <mi>C</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> may be identified with a point to the upper left, as they have the same dynamics. To illustrate this, four points <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> </mrow> </semantics></math> are shown on <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>L</mi> <mo>)</mo> </mrow> </semantics></math>, with their preimages under <span class="html-italic">F</span> marked on <span class="html-italic">L</span>. For instance, the two points marked <math display="inline"><semantics> <mrow> <msup> <mi>F</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> both map to <span class="html-italic">A</span> and share the same future thereafter. The values used in this graph are <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </semantics></math>. Lemmas 1, 2, and Theorem 2 have shown that this depiction is typical for all lines with this slope for any <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msup> <mi>σ</mi> <mn>2</mn> </msup> <mo>∈</mo> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>The region <span class="html-italic">U</span> bounded by <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and the curve <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>C</mi> </mrow> </semantics></math>, where the parameters values are <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </semantics></math>. This region is forward invariant, globally attracting, and the restriction of <span class="html-italic">F</span> to <span class="html-italic">U</span> is injective, as proved in Theorem 3. Lemma 2 shows that the general shape of region <span class="html-italic">U</span> is typical.</p> Full article ">Figure 4
<p>The figure depicts a generic representation of the regions <math display="inline"><semantics> <msub> <mi>R</mi> <mi>i</mi> </msub> </semantics></math> as determined by the isoclines and the curve <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>C</mi> </mrow> </semantics></math>. In the plot shown, the parameter values are <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </semantics></math>, though Lemma 2, and the equations for the isoclines show that this plot is typical for all <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>∈</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>A prototype of the stability region in the parameter space of the interior fixed point <math display="inline"><semantics> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>∗</mo> </msup> <mo>,</mo> <msup> <mi>x</mi> <mo>∗</mo> </msup> <mo>)</mo> </mrow> </semantics></math> of Model (<a href="#FD3-symmetry-16-01139" class="html-disp-formula">3</a>).</p> Full article ">Figure 6
<p>Numerically computed the unstable manifolds of the origin for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </semantics></math>. The <math display="inline"><semantics> <mi>β</mi> </semantics></math> values in the four plots are <math display="inline"><semantics> <mrow> <mn>0.1</mn> </mrow> </semantics></math>, 1, 20, and 100, respectively. The fixed point <math display="inline"><semantics> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>∗</mo> </msup> <mo>,</mo> <msup> <mi>x</mi> <mo>∗</mo> </msup> <mo>)</mo> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </semantics></math> is locally asymptotically stable for <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>&lt;</mo> <mn>65</mn> </mrow> </semantics></math>. In the first three images, <math display="inline"><semantics> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>∗</mo> </msup> <mo>,</mo> <msup> <mi>x</mi> <mo>∗</mo> </msup> <mo>)</mo> </mrow> </semantics></math> is locally asymptotically stable and the unstable manifold <math display="inline"><semantics> <mrow> <msup> <mi>W</mi> <mi>u</mi> </msup> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> is asymptotic to the fixed point <math display="inline"><semantics> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>∗</mo> </msup> <mo>,</mo> <msup> <mi>x</mi> <mo>∗</mo> </msup> <mo>)</mo> </mrow> </semantics></math>. For small <math display="inline"><semantics> <mi>β</mi> </semantics></math>, it converges directly to it, while for larger <math display="inline"><semantics> <mi>β</mi> </semantics></math> it spirals into it. In the last image, these spirals converge to an invariant curve instead of to the unstable fixed point.</p> Full article ">Figure 7
<p>Numerically computed unstable manifold <math display="inline"><semantics> <mrow> <msup> <mi>W</mi> <mi>u</mi> </msup> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mstyle> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </semantics></math>, along with lines parallel to the eigenvectors at <math display="inline"><semantics> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>∗</mo> </msup> <mo>,</mo> <msup> <mi>x</mi> <mo>∗</mo> </msup> <mo>)</mo> </mrow> </semantics></math> (shown as dotted lines). The isoclines are also shown.</p> Full article ">
18 pages, 697 KiB  
Article
Improving Influenza Epidemiological Models under Caputo Fractional-Order Calculus
by Nahaa E. Alsubaie, Fathelrhman EL Guma, Kaouther Boulehmi, Naseam Al-kuleab and Mohamed A. Abdoon
Symmetry 2024, 16(7), 929; https://doi.org/10.3390/sym16070929 - 20 Jul 2024
Cited by 3 | Viewed by 724
Abstract
The Caputo fractional-order differential operator is used in epidemiological models, but its accuracy benefits are typically ignored. We validated the suggested fractional epidemiological seasonal influenza model of the SVEIHR type to demonstrate the Caputo operator’s relevance. We analysed the model using fractional calculus, [...] Read more.
The Caputo fractional-order differential operator is used in epidemiological models, but its accuracy benefits are typically ignored. We validated the suggested fractional epidemiological seasonal influenza model of the SVEIHR type to demonstrate the Caputo operator’s relevance. We analysed the model using fractional calculus, revealing its basic properties and enhancing our understanding of disease progression. Furthermore, the positivity, bounds, and symmetry of the numerical scheme were examined. Adjusting the Caputo fractional-order parameter α = 0.99 provided the best fit for epidemiological data on infection rates. We compared the suggested model with the Caputo fractional-order system and the integer-order equivalent model. The fractional-order model had lower absolute mean errors, suggesting that it could better represent sickness transmission and development. The results underline the relevance of using the Caputo fractional-order operator to improve epidemiological models’ precision and forecasting. Integrating fractional calculus within the framework of symmetry helps us build more reliable models that improve public health interventions and policies. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Models)
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Figure 1

Figure 1
<p>Transmission dynamics of the fractional influenza model.</p> Full article ">Figure 2
<p>Comparing the classical model (<math display="inline"><semantics> <mi mathvariant="sans-serif">α</mi> </semantics></math> = 1) to the Caputo model (<math display="inline"><semantics> <mi mathvariant="sans-serif">α</mi> </semantics></math> = 0.99) using real data.</p> Full article ">Figure 3
<p>The dynamical behaviour of the susceptible individuals included in fractional model (<a href="#FD4-symmetry-16-00929" class="html-disp-formula">4</a>).</p> Full article ">Figure 4
<p>The dynamical behaviour of the vaccinated individuals included in fractional model (<a href="#FD4-symmetry-16-00929" class="html-disp-formula">4</a>).</p> Full article ">Figure 5
<p>The dynamical behaviour of the exposed individuals included in fractional model (<a href="#FD4-symmetry-16-00929" class="html-disp-formula">4</a>).</p> Full article ">Figure 6
<p>The dynamical behaviour of the hospitalized individuals included in fractional model (<a href="#FD4-symmetry-16-00929" class="html-disp-formula">4</a>).</p> Full article ">Figure 7
<p>The dynamical behaviour of the recovered individuals included in fractional model (<a href="#FD4-symmetry-16-00929" class="html-disp-formula">4</a>).</p> Full article ">Figure 8
<p>The effects of memory trace on each population of model (<a href="#FD4-symmetry-16-00929" class="html-disp-formula">4</a>). (<b>a</b>) S population, (<b>b</b>) V population, (<b>c</b>) E population, (<b>d</b>) I population, (<b>e</b>) H population, (<b>f</b>) R population.</p> Full article ">
18 pages, 313 KiB  
Article
Partially Nonclassical Method and Conformal Invariance in the Context of the Lie Group Method
by Georgy I. Burde
Symmetry 2024, 16(7), 875; https://doi.org/10.3390/sym16070875 - 10 Jul 2024
Viewed by 832
Abstract
The basic idea of the ‘partially nonclassical method’, developed in the present paper, is to apply the invariance requirement of the Lie group method using not all differential consequences of the invariant surface condition but only part of them. It differs from the [...] Read more.
The basic idea of the ‘partially nonclassical method’, developed in the present paper, is to apply the invariance requirement of the Lie group method using not all differential consequences of the invariant surface condition but only part of them. It differs from the ‘classical’ method, in which the invariant surface condition is not used, and from the ‘nonclassical’ method, in which all the differential consequences are used. It provides additional possibilities for the symmetry analysis of partial differential equations (PDEs), as compared with the ‘classical’ and ‘nonclassical’ methods, in the so-named no-go case when the group generator, associated with one of the independent variables, is identically zero. The method is applied to the flat steady-state boundary layer problem, reduced to an equation for the stream function, and it is found that applying the partially nonclassical method in the no-go case yields new symmetry reductions and new exact solutions of the boundary layer equations. A computationally convenient unified framework for the classical, nonclassical and partially nonclassical methods (λ-formulation) is developed. The issue of conformal invariance in the context of the Lie group method is considered, stemming from the observation that the classical Lie method procedure yields transformations not leaving the differential polynomial of the PDE invariant but modifying it by a conformal factor. The physical contexts, in which that observation could be important, are discussed using the derivation of the Lorentz transformations of special relativity as an example. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Models)
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Figure 1

Figure 1
<p>Profiles of the longitudinal velocity at different cross-sections for the flow (<a href="#FD58-symmetry-16-00875" class="html-disp-formula">58</a>), k = 1.</p> Full article ">
11 pages, 282 KiB  
Article
Diagonals–Parameter Symmetry Model and Its Property for Square Contingency Tables with Ordinal Categories
by Kouji Tahata and Kohei Matsuda
Symmetry 2024, 16(6), 768; https://doi.org/10.3390/sym16060768 - 19 Jun 2024
Viewed by 831
Abstract
The diagonals–parameter symmetry (DPS) model is a proposed method for analyzing square contingency tables with ordinal categories. Previously, it was stated that the generalized DPS (DPS[f]) model was equivalent to the DPS model for any function f, but the proof [...] Read more.
The diagonals–parameter symmetry (DPS) model is a proposed method for analyzing square contingency tables with ordinal categories. Previously, it was stated that the generalized DPS (DPS[f]) model was equivalent to the DPS model for any function f, but the proof was not provided. This paper presents the derivation of the DPS[f] model and the proof of the relationship between the two models. The findings offer various interpretations of the DPS model. Additionally, a new model is considered, and it is shown that the proposed model and the DPS[f] model are separable. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Models)
9 pages, 892 KiB  
Article
Symmetric U-Net Model Tuned by FOX Metaheuristic Algorithm for Global Prediction of High Aerosol Concentrations
by Dušan P. Nikezić, Dušan S. Radivojević, Nikola S. Mirkov, Ivan M. Lazović and Tatjana A. Miljojčić
Symmetry 2024, 16(5), 525; https://doi.org/10.3390/sym16050525 - 26 Apr 2024
Viewed by 726
Abstract
In this study, the idea of using a fully symmetric U-Net deep learning model for forecasting a segmented image of high global aerosol concentrations is implemented. As the forecast relies on historical data, the model used a sequence of the last eight segmented [...] Read more.
In this study, the idea of using a fully symmetric U-Net deep learning model for forecasting a segmented image of high global aerosol concentrations is implemented. As the forecast relies on historical data, the model used a sequence of the last eight segmented images to make the prediction. For this, the classic U-Net model was modified to use ConvLSTM2D layers with MaxPooling3D and UpSampling3D layers. In order to achieve complete symmetry, the output data are given in the form of a series of eight segmented images shifted by one image in the time sequence so that the last image actually represents the forecast of the next image of high aerosol concentrations. The proposed model structure was tuned by the new FOX metaheuristic algorithm. Based on our analysis, we found that this algorithm is suitable for tuning deep learning models considering their stochastic nature. It was also found that this algorithm spends the most time in areas close to the optimal value where there is a weaker linear correlation with the required metric and vice versa. Taking into account the characteristics of the used database, we concluded that the model is capable of generating adequate data and finding patterns in the time domain based on the ddc and dtc criteria. By comparing the achieved results of this model using the AUC-PR metric with the previous results of the ResNet3D-101 model with transfer learning, we concluded that the proposed symmetric U-Net model generates data better and is more capable of finding patterns in the time domain. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Models)
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Figure 1

Figure 1
<p>Proposed structure of the symmetric U-Net architecture.</p> Full article ">Figure 2
<p>Distribution of parameters searched by the FOX algorithm.</p> Full article ">
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