A holistic physics-informed neural network solution for precise destruction of breast tumors using focused ultrasound on a realistic breast model by Salman Lari, Hossein Rajabzadeh, Mohammad Kohandel, Hyock Ju Kwon

“…Additionally, employing PINN for estimating partial differential equations (PDE) solutions can notably decrease the enormous number of discretized elements needed. …”
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Melting ternary hybrid nanofluid stagnation point flow with velocity slip past a stretching/shrinking sheet: Numerical simulation and validation via P2SATRA by Nur Syahirah Wahid, Nur Ezlin Zamri, Siti Zulaikha Mohd Jamaludin, Nur Hazirah Adilla Norzawary, Mohd Shareduwan Mohd Kasihmuddin, Mohd. Asyraf Mansor, Norihan Md Arifin, Ioan Pop

“…The governing partial differential equations (PDEs) are initially formulated and subsequently reduced to ordinary differential equations (ODEs). …”
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Magnetohydrodynamic Peristaltic Propulsion of Casson Nanofluids With Slip Effects Over Heterogeneous Rough Channel by Hanumesh Vaidya, Fateh Mebarek‐Oudina, Rakesh Kumar, C. Rajashekhar, Kerehalli Vinayaka Prasad, Sangeeta Kalal, Kottakkaran Sooppy Nisar

“…A novel rough non‐uniform model is effectively governed by a set of nonlinear coupled governing partial differential equations, which are simplified under long wavelength and creeping flow approximations. …”
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Extrapolation of cavitation and hydrodynamic pressure in lubricated contacts: a physics-informed neural network approach by Faras Brumand-Poor, Freddy Kokou Azanledji, Nils Plückhahn, Florian Barlog, Lukas Boden, Katharina Schmitz

“…In particular, physics-informed neural networks (PINNs) are as effective hybrid solvers that combine data-driven and physics-based methods to solve the partial differential equations that drive EHL simulations. By integrating physical laws into the parameter optimization of the neural network (NN), PINNs provide accurate and fast solutions. …”
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Accelerating phase field simulations through a hybrid adaptive Fourier neural operator with U-net backbone by Christophe Bonneville, Nathan Bieberdorf, Arun Hegde, Mark Asta, Habib N. Najm, Laurent Capolungo, Cosmin Safta

“…However, the LMD governing equations in these models often involve coupled non-linear partial differential equations (PDE), which are challenging to solve numerically. …”
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Numerical Analysis on the Storage of Nuclear Waste in Gas-Saturated Deep Coal Seam by Teng Teng, Yuming Wang, Xiaoyan Zhu, Xiangyang Zhang, Sihai Yi, Guowei Fan, Bin Liu

“…As the second step, a finite element numerical model and numerical simulation are developed to analyze the storage of nuclear waste in a gas-saturated deep coal seam based on the partial differential equations (PDE) solver of COMSOL Multiphysics with MATLAB. …”
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Heat transportation of 3D chemically reactive flow of Jeffrey nanofluid over a porous frame with variable thermal conductivity by Nahid Fatima, Aaqib Majeed, Taoufik Saidani, Nouman Ijaz, Kamal Barghout, Nidal Abu-Libdeh

“…To model these phenomena, we employ the boundary layer approximation to derive a system of partial differential equations (PDEs). These PDEs are subsequently simplified into more manageable ordinary differential equations (ODEs) using the similarity variables. …”
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Application of artificial intelligence brain structure-based paradigm to predict the slip condition impact on magnetized thermal Casson viscoplastic fluid model under combined temp... by Umar Farooq, Shan Ali Khan, Haihu Liu, Muhammad Imran, Lotfi Ben Said, Aleena Ramzan, Taseer Muhammad

“…With the help of a similarity transformation, the complex partial differential equations governing the flow and energy take on the form of nonlinear ordinary differential equations. …”
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Second order slip micropolar MHD hybrid nanofluid flow over a stretching surface with uniform heat source and activation energy: Numerical computational approach by Syed Arshad Abas, Hakeem Ullah, Mehreen Fiza, Ali Akgul, Aasim Ullah Jan, Magda Abd El-Rahman, Seham M. Al-Mekhlafi

“…Methodology: A mathematical model is formulated based on boundary-layer approximations, leading to a system of partial differential equations (PDEs) that describe the flow, thermal, and concentration fields. …”
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A multi-layer neural network-based evaluation of MHD radiative heat transfer in Eyring–Powell fluid model by Muflih Alhazmi, Zahoor Shah, Muhammad Asif Zahoor Raja, Nafisa A. Albasheir, Maryam Jawaid

“…To facilitate analysis, governing partial differential equations (PDEs) are converted into ordinary differential equations (ODEs) only with the aid of similarity transformations. …”
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Non Linear Thermal Radiation Analysis of Electromagnetic Chemically Reacting Ternary Nanofluid Flow over a Bilinear Stretching Surface by Shobha V, Hasan Mulki, Baskar P, S. Suresh Kumar Raju, Saleh Mahmoud, Mostafa Abdrabboh, S.V.K. Varma

“…Methodology: The governing nonlinear partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) using similarity transformations. …”
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ANN-based two hidden layers computational procedure for analysis of heat transport dynamics in polymer-based trihybrid Carreau nanofluid flow over needle geometry by Adil Darvesh, Fethi Mohamed Maiz, Basma Souayeh, Luis Jaime Collantes Santisteban, Hakim AL. Garalleh, Afnan Al Agha, Lucerito Katherine Ortiz García, Nicole Anarella Sánchez-Miranda

“…Methodology: The physical model is originally formed with the help of partial differential equations (PDEs), that are formulated with pre-defined assumption of fluid flow mechanism. …”
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Mathematical modeling in autoimmune diseases: from theory to clinical application by Yaroslav Ugolkov, Yaroslav Ugolkov, Yaroslav Ugolkov, Antonina Nikitich, Antonina Nikitich, Cristina Leon, Gabriel Helmlinger, Kirill Peskov, Kirill Peskov, Kirill Peskov, Kirill Peskov, Victor Sokolov, Victor Sokolov, Alina Volkova, Alina Volkova

“…. ≥70% of the models were developed as nonlinear systems of ordinary differential equations, others – as partial differential equations, integro-differential equations, Boolean networks, or probabilistic models. …”
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Role of stability analysis and waste discharge concentration of ternary hybrid nanofluid in a non-Newtonian model with slip boundary conditions by Nurhana Mohamad, Shuguang Li, Umair Khan, Anuar Ishak, Ali Elrashidi, Mohammed Zakarya

“…The suitable similarity transformations are utilized to transform the partial differential equations (PDEs) into ordinary differential equations (ODEs). …”
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Permeability Models for Magma Flow through the Earth's Mantle: A Lie Group Analysis by N. Mindu, D. P. Mason

“…The migration of melt through the mantle of the Earth is governed by a third-order nonlinear partial differential equation for the voidage or volume fraction of melt. …”
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Asian Option Pricing with Transaction Costs and Dividends under the Fractional Brownian Motion Model by Yan Zhang, Di Pan, Sheng-Wu Zhou, Miao Han

“…The pricing problem of geometric average Asian option under fractional Brownian motion is studied in this paper. The partial differential equation satisfied by the option’s value is presented on the basis of no-arbitrage principle and fractional formula. …”
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Recognition of High Difference Features in Urban Planning Images Based on Morphological Filtering by Peipei Liu

“…This paper mainly studies the image enhancement method based on partial differential equation. By analysing the combination of partial differential equation theory and enhancement, aiming at the shortcomings of low recognition accuracy, high error rate, and long time consuming in the current method of urban planning image feature recognition, a feature enhancement and simulation of urban planning image based on partial differential equation method is proposed; the preprocessing of urban planning image is realized by collecting the urban planning image. …”
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Accuracy, Robustness, and Efficiency of the Linear Boundary Condition for the Black-Scholes Equations by Darae Jeong, Seungsuk Seo, Hyeongseok Hwang, Dongsun Lee, Yongho Choi, Junseok Kim

“…We briefly review and investigate the performance of various boundary conditions such as Dirichlet, Neumann, linear, and partial differential equation boundary conditions for the numerical solutions of the Black-Scholes partial differential equation. …”
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Viscosity Solution of Mean-Variance Portfolio Selection of a Jump Markov Process with No-Shorting Constraints by Moussa Kounta

“…In this situation the Hamilton-Jacobi-Bellman (HJB) equation of the value function of the auxiliary problem becomes a coupled system of backward stochastic partial differential equation. In fact, the value function V often does not have the smoothness properties needed to interpret it as a solution to the dynamic programming partial differential equation in the usual (classical) sense; however, in such cases V can be interpreted as a viscosity solution. …”
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An Analytic Solution for a Vasicek Interest Rate Convertible Bond Model by A. S. Deakin, Matt Davison

“…This paper provides the analytic solution to the partial differential equation for the value of a convertible bond. …”
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