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Mathematics
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Advances in PDE-Based Models, Artificial Intelligence and Deep Learning
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Advances in PDE-Based Models, Artificial Intelligence and Deep Learning

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: 30 June 2024 | Viewed by 4227

Special Issue Editor

Prof. Dr. Tudor Barbu

E-Mail Website
Guest Editor
Institute of Computer Science of the Romanian Academy, Iași Branch, 700481 lasi, Romania
Interests: image processing and analysis; computer vision machine learning; partial differential equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Partial differential equations (PDE), which have long been used to formulate dynamical phenomena, have been applied successfully in various digital image processing and analysis and computer vision fields, such as image filtering, inpainting, segmentation, compression, decomposition, registration, and optical flow estimation, in the last 35 years. They have also been applied in artificial intelligence, where PDE-based models have been used to generate artificial neural network architectures.

An important sub-domain of artificial intelligence, deep learning, also represents an application area of partial differential equations. Deep learning represents a family of advanced machine learning techniques based on artificial neural networks with representation learning, such as deep belief networks, deep reinforcement learning, recurrent neural networks, and convolutional neural networks. Deep neural networks, which have been applied successfully to all image processing and analysis and computer vision fields, have many connections to PDE models. Thus, neural partial differential equations (NPDE) can be used to describe AI systems and the architectures of some deep networks that can be interpreted as nonlinear PDEs, such as recurrent neural networks (RNN). Further, evolution PDEs can be learned from given datasets using deep neural networks, which also predict their dynamical behavior. Convolutional neural networks (CNN) have been increasingly used to numerically solve PDE-based models, providing fast and effective numerical approximations for them. Finally, PDE and deep-learning-based models can be combined successfully to provide more effective image and video processing and computer vision solutions. 

This proposed Special Issue deals with all these mentioned connections between PDE models, AI systems, and deep learning, especially those related to image analysis and computer vision, the main purpose being the dissemination of advanced and original research in these scientific domains and bringing together of researchers working in partial differential equations, image processing and analysis, computer vision, artificial intelligence, and deep learning, so as to extend the existing knowledge in these important fields.

We thus encourage you to send high-quality papers that disseminate new research achievements for this issue.

Prof. Dr. Tudor Barbu
Guest Editor

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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • partial differential equations
  • image denoising and restoration
  • image interpolation
  • image segmentation
  • image registration
  • image compression
  • computer vision
  • artificial intelligence
  • deep learning
  • neural partial differential equations
  • PDE-based AI system description
  • CNN-based numerical PDE solving
  • PDE-based deep network interpretation
  • DL-based PDE model learning

Published Papers (2 papers)

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Research

12 pages, 1642 KiB  
Article
CNN-Based Temporal Video Segmentation Using a Nonlinear Hyperbolic PDE-Based Multi-Scale Analysis
by Tudor Barbu
Mathematics 2023, 11(1), 245; https://doi.org/10.3390/math11010245 - 3 Jan 2023
Cited by 2 | Viewed by 1299
Abstract
An automatic temporal video segmentation framework is introduced in this article. The proposed cut detection technique performs a high-level feature extraction on the video frames, by applying a multi-scale image analysis approach combining nonlinear partial differential equations (PDE) to convolutional neural networks (CNN). [...] Read more.
An automatic temporal video segmentation framework is introduced in this article. The proposed cut detection technique performs a high-level feature extraction on the video frames, by applying a multi-scale image analysis approach combining nonlinear partial differential equations (PDE) to convolutional neural networks (CNN). A nonlinear second-order hyperbolic PDE model is proposed and its well-posedness is then investigated rigorously here. Its weak and unique solution is determined numerically applying a finite difference method-based numerical approximation algorithm that quickly converges to it. A scale-space representation is then created using that iterative discretization scheme. A CNN-based feature extraction is performed at each scale and the feature vectors obtained at multiple scales are concatenated into a final frame descriptor. The feature vector distance values between any two successive frames are then determined and the video transitions are identified next, by applying an automatic clustering scheme on these values. The proposed PDE model, its mathematical investigation and discretization, and the multi-scale analysis based on it represent the major contributions of this work. Some temporal segmentation experiments and method comparisons that illustrate the effectiveness of the proposed framework are finally described in this research paper. Full article
(This article belongs to the Special Issue Advances in PDE-Based Models, Artificial Intelligence and Deep Learning)
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12 pages, 537 KiB  
Article
Alternative Method to Estimate the Fourier Expansions and Its Rate of Change
by Johnny Rodríguez-Maldonado, Cornelio Posadas-Castillo and Ernesto Zambrano-Serrano
Mathematics 2022, 10(20), 3832; https://doi.org/10.3390/math10203832 - 17 Oct 2022
Cited by 1 | Viewed by 1192
Abstract
This paper presents a methodology to obtain the Fourier coefficients (FCs) and the derivative Fourier coefficients (DFCs) from an input signal. Based on the Taylor series that approximates the input signal into a trigonometric signal model through the Kalman filter, consequently, the signal’s [...] Read more.
This paper presents a methodology to obtain the Fourier coefficients (FCs) and the derivative Fourier coefficients (DFCs) from an input signal. Based on the Taylor series that approximates the input signal into a trigonometric signal model through the Kalman filter, consequently, the signal’s and successive derivatives’ coefficients are obtained with the state prediction and the state matrix inverse. Compared to discrete Fourier transform (DFT), the new class of filters provides noise reduction and sidelobe suppression advantages. Additionally, the proposed Taylor–Kalman–Fourier algorithm (TKFA) achieves a null-flat frequency response around the frequency operation. Moreover, with the proposed TKFA method, the decrement in the inter-harmonic amplitude is more significant than that obtained with the Kalman–Fourier algorithm (KFA), and the neighborhood of the null-flat frequency is expanded. Finally, the approximation of the input signal and its derivative can be performed with a sum of functions related to the estimated coefficients and their respective harmonics. Full article
(This article belongs to the Special Issue Advances in PDE-Based Models, Artificial Intelligence and Deep Learning)
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