PDE-GANet: Partial differential equation discovery powered by adversarial learning.

Physics-informed neural networks in water and wastewater systems: a critical review.

Restoration of partially damaged fingerprints using a partial differential equation

Translating in vitro buccal permeation to in vivo and whole‑body exposure using in silico cell‑based and physiologically-based pharmacokinetic modelling.

Variable step-size IMEX scheme for a partial differential equation with delays and mixed derivative from option pricing under hard-to-borrow model

B cells are important components of the adaptive immune system, responsible for antibody production and working as antigen-presenting cells. B cells display protein receptors on their membrane, which bind with foreign antigens and process them before presenting them to T cells. In this work, we present a stochastic process modeling the dynamics of such receptors on the B cell. The model consists of a two-dimensional birth-death process {(X(t),Y(t)),t≥0} having linear transition rates, where X(t) and Y(t) represent the number of free and occupied receptors, respectively. After determining the partial differential equation for the probability generating function of the process, we compute the main moments of the process, including the covariance. The transient and asymptotic behavior of the means of X(t) and Y(t) is also studied. Throughout the paper, we provide insights into the biological significance of each parameter on the system's dynamics. In addition, we conduct a sensitivity analysis to assess how variations in the model parameters affect the first-order moments. Such analysis shows that minimal variations of the parameters representing the binding frequency of antigens and B-cell receptors, when happening in the initial instants of the process, result in noticeable alterations of the number of occupied receptors.

An Isogeometric Tearing and Interconnecting (IETI) method for solving high order partial differential equations over planar multi-patch geometries

MBNO: Mamba-based neural operators for solving partial differential equations

Accurate solution of high-dimensional partial differential equations: Research on data-physics collaborative modeling and adaptive sampling

• A Geometry-Informed Neural Operator Transformer (GINOT) is proposed for forward predictions on arbitrary geometries. • GINOT encodes surface point clouds that are unordered, have non-uniform point density, and varying numbers of points. • GINOT effectively processes complex, arbitrary geometries and varying input conditions with good predictive accuracy. Machine-learning-based surrogate models offer significant computational efficiency and faster simulations compared to traditional numerical methods, especially for problems requiring repeated evaluations of partial differential equations. This work introduces the Geometry-Informed Neural Operator Transformer (GINOT), which integrates the transformer architecture with the neural operator framework to enable forward predictions on arbitrary geometries. GINOT employs a sampling and grouping strategy together with an attention mechanism to encode surface point clouds that are unordered, exhibit non-uniform point densities, and contain varying numbers of points for different geometries. The geometry information is seamlessly integrated with query points in the solution decoder through the attention mechanism. The performance of GINOT is validated on multiple challenging datasets, showcasing its accuracy and generalization capabilities for complex and arbitrary 2D and 3D geometries.

We introduce finite-element Gaussian processes (FEGPs), a novel physics-informed machine learning approach for solving inverse problems involving steady-state, linear partial differential equations (PDEs). Our framework combines a Gaussian process prior for the unknown solution function with a likelihood that incorporates the PDE in its weak form, using a finite-element approximation. This approach offers significantly better scalability than physics-informed Gaussian processes (PIGPs), which rely on the strong form of the PDE. Through numerical experiments on a range of synthetic benchmark problems, we show that FEGPs offer results which outperform PIGPs, and are competitive with physics-informed neural networks (PINNs) with improved uncertainty quantification.

This paper is an in-depth examination of split points in Musielak-Orlicz spaces LΦ , giving a full description of their geometry, and answering various outstanding questions about the geometry of these spaces. The discussion is based on the classical duality arguments together with the recent concepts of the theory of non-uniformly convex spaces and variable-exponent analysis. The key achievements of the paper may be concluded as follows. To begin with, we create conditions which are such as to ensure that split points exist. They coincide with the strict convexity of the modular function Φ(x,·), the Δ₂-condition of validity of Φ and the conjugated version, along with some assumptions of spatial symmetry. Second, we obtain a blame-sharing characterization in the form of introducing a correction term D(x,g). This expression is a generalization of the popular system of Giles that includes ideas of the modular fixed-point theory. Third, we examine the computational factors of determining split points. We find that at L(p) > 1 the instability index is computationally difficult. Specifically, we explain the cases where the identification process is unsuccessful in the Lebesgue spaces of variable-exponent Lp(x). Lastly, we demonstrate the applicability of the theory to an application to variable-exponent partial differential equations. Specifically, we consider the use of the property of split points in the norm-attainment of solutions of the equation: -Δ p(x) u = f These results imply a number of possible extensions, such as additional research studies in the context of noncommutative Musielak-Orlicz spaces.

This paper presents a comprehensive analysis of magnetohydrodynamic (MHD) flow, radiative heat transfer, and mass transport in nanofluids interacting with an incompressible, electrically conducting fluid. The study accounts for the combined effects of Joule heating, viscous dissipation, and a first-order chemical reaction over a porous plate embedded in a porous medium subjected to a prescribed heat flux. A numerical investigation is performed on the boundary-layer flow model involving three distinct nanoparticle types Cu , A l 2 O 3 and Ag . The governing equations for momentum, energy, and species concentration are formulated under the boundary-layer approximation. By employing similarity transformations, the coupled nonlinear partial differential equations with associated boundary conditions are reduced to a system of ordinary differential equations (ODEs) defined over a semi-infinite domain. This system is solved numerically using a hybrid scheme that integrates the Shooting technique with the fourth-order Adams–Moulton method. The accuracy of the results is confirmed through comparison with previously published data, demonstrating excellent agreement. The effects of key physical parameters on the velocity, temperature, and concentration fields are examined and illustrated through graphical and tabular analyses. Furthermore, thermophysical property correlations are provided. The findings indicate that an increase in nanoparticle volume fraction leads to elevated temperature profiles, thereby enhancing the Schmidt number. Thus, the results provide a clear understanding of fluid flow behaviour in applications such as nuclear reactor cooling systems, polymer extrusion processes, and electromagnetic magnetohydrodynamic generators.

Stochastic partial differential equations (SPDEs) have become key tools for modeling randomly disturbed systems in a broad spectrum of scientific and engineering disciplines. Among them, the optical soliton propagation in birefringent fibers has been of great interest because of the natural stochastic effects present in real optical communication systems. Although some research has been conducted on soliton dynamics in deterministic and integer-order models, the literature has few studies focusing on fractional stochastic frameworks coupled with M-truncated fractional operators. This paper fills this gap by proposing and studying the fractional stochastic Biswas–Arshed equation (FSBAE) that includes multiplicative white noise to represent random perturbations during optical signal propagation. We employ the [Formula: see text] method with Itô calculus and M-truncated fractional derivatives: in order to arrive at three different classes of solutions: stochastic optical breather solitons, M-shaped solitons, and singular solitons. The first contribution of this work consists in the combination of M-truncated fractional calculus — a not extensively developed operator in stochastic nonlinear optics — within the FSBAE, for the first time applied to this model. Comparative graphical analysis under conditions of different levels of white noise and fractional orders also identifies the robustness and stability of the solutions in close proximity to the zero-noise limit. This research adds to the development of soliton theory in that it has proposed a new mathematical model that can capture both nonlocal memory effect and stochastic variability, which are essential in real optical systems. The efficiency, simplicity, and flexibility of the approach make it an effective tool in solving a wider class of nonlinear stochastic issues in optical engineering and other fields dealing with stochastic partial differential equations.

Abstract Physics-Informed Neural Networks are deep learning neural networks explicitly conceived as an alternative solver of partial differential equations with respect to standard numerical techniques. PINNs offer some unique features, such as the capability of constraining the solution with internal or external and local or integral information, allowing to take into account uncertainty of this information. They can also be constrained with incomplete physics equations, allowing the development of modelling tools. Therefore, they offer the possibility of developing a unique framework, which permits to combine physics and data. In this work, their potential has been investigated by applying them to one of the most important inverse problems in tokamaks, the plasma equilibrium reconstruction. More specifically, an advanced PINN-based equilibrium reconstruction method has been developed that combines multi-diagnostic constraints with high-fidelity physics modelling of the measurements, able to take into account both non-linearities and relativistic effects. All the relevant diagnostics have been included in the study, confirming the potential of the technology to perform also integrated data analysis. A series of numerical tests, performed with the help of the Tokalab platform, have proven the quality of the results in cases, for which the right solution is known. After this validation, the developed tools have been applied to analyse various JET discharges, with particular attention to high performance experiments in DT. A detailed comparison with the reference inversion codes used on JET (EFIT, EFTP and EFTF) is reported together with diagnostic ablation tests, confirming both the accuracy and the reliability of the approach. The obtained performances motivate various future developments such as the implementation of multi-fluid MHD equations, plasma dynamics reconstruction, and acceleration schemes to reduce the computational times.

The proposed work focuses on study of homotopy based approximation techniques for evaluating partial differential equations (PDEs) with continuous and piece-wise initial conditions. The well proposed homotopy analysis method (HAM) is taken into consideration. The approximate method based on a zero-order deformation equations in topology contains the auxiliary operator for mapping an initial estimation to the unknown solution and ensure rapid convergence of the given series approximate solution. Implementation of residual error through algebraic equations depicts that the proposed zero-order deformation equation essentially increase the rate of the convergence region and series solution and concede greater freedom in the choosing of convergence operators than the traditional approximate methods. As a starting point of approach, we applied HAM to linear system of PDEs in first order form with prescribed conditions including presence and absence of governing parameters. The convergence of system of PDEs is analysed through auxiliary parameter value. The derived series solution obtained from HAM is compared with exact solution in order to confirm accuracy and effectiveness. The HAM shows rapid convergence than the usual numerical and approximate approach and confirm accuracy in comparison with exact solution in limiting case.

The local thermal non-equilibrium (LTNE) effects are used for temperature variances between liquid and solid phases in porous media and heat transfer systems, such as nuclear waste disposal, geothermal reservoirs and packed bed reactors. This study provides a comprehensive modeling of hybrid nanofluid (HNF) flow along a Riga plate, combining impacts of LTNE between the solid and liquid stages. LTNE is used to determine the energy equations and distinct temperature profiles for the liquid and solid phases. The model considers a Riga plate fixed in a Darcy–Forchheimer porous medium to highlight the dominant electrohydrodynamic interactions. The effect of activation energy is also considered. The governing partial differential equations (PDEs) are changed into ordinary differential equations (ODEs) by the application of similarity transformations. The bvp-4c technique with a shooting methodology is used to describe the created system of ODEs. The impacts of important parameters on concerned profiles have been demonstrated in tabular and graphical forms. The outputs determine that LTNE significantly improves thermal performance, while the Riga plate efficiently moderates boundary layer dynamics. The temperature for solid phase decreases by raising the parameter of non-dimensional inter phase heat transfer. The concentration profile declines by growing the Lewis number. The Darcy–Forchheimer effects significantly influence flow and thermal transport. The study offers understanding of improving HNF applications for thermal management.

The generalized fractional Kundu-Mukherjee-Naskar equation (gFKMNE) is a nonlinear fractional partial differential equation (NFPDE) that models nonlinear pulse transmission in communication systems and optical fibers. This work investigates the gFKMNE in (2+1) dimensions, seeking optical soliton solutions. Using a wave transformation, the gFKMNE is converted into nonlinear ordinary differential equations (NODEs) of integer order. The modified extended direct algebraic approach is then applied to solve the NODE, yielding nonlinear algebraic equations and series-form solutions. The solutions, obtained using Maple-13, reveal optical soliton solutions for the gFKMNE. These soliton solutions can be stacked to produce black lattices in optical media, visible in contour plots and 3D photographs. These dark soliton lattices are crucial in telecommunications, optical signal processing, and nonlinear optics. Further analysis examines the influence of various parameters on soliton behavior, demonstrating that soliton solutions exhibit distinct features depending on parameters like amplitude, width, and velocity. This study showcases the effectiveness of the modified extended direct algebraic technique in solving complex NFPDEs, providing new insights into soliton behavior in optical media.

This paper assesses the stability results for partial differential equations by means of the comparison principle. By using the python software, numerical example is given to illustrates the rapid convergence of the system solution which goes a long way to show the effectiveness of the adopted approach.

Taylor dispersion of a solute in a pulsatile flow of a viscoelastic fluid, whose constitutive equation follows the Maxwell model, through an eccentric annulus is investigated in this work. To determine the effective dispersion coefficient, $\mathscr{D}_{\textit{eff}}$ , we have used the multiple-scale analysis in conjunction with the homogenization method. The governing equation describing this dispersive phenomenon for solute concentration is the advection-diffusion equation, which depends on the velocity profile. Therefore, the momentum equation must be solved in advance. A hyperbolic partial differential equation in a bipolar coordinate system was derived by combining the Cauchy momentum equation with Maxwell’s constitutive equation. Parameters such as the Womersley number, ${\textit{Wo}}$ , and the Deborah number, ${\textit{De}}$ , control the time-dependent flow and viscoelasticity, respectively. For low Womersley numbers, i.e. for low frequencies, an increase in the Deborah number, the eccentricity, $\phi$ , and gap width, $\gamma$ , leads to an enhancement of the effective dispersion coefficient. For instance, a fluid with ${\textit{De}} = 5$ could increase $\mathscr{D}_{\textit{eff}}$ by two orders of magnitude compared with a Newtonian fluid with the same settings ( $\phi = 0.3$ and ${\textit{Wo}} = 0.1$ ). However, this enhancement due to the viscoelastic effect is only significant at low frequencies. An advection-diffusion equation for the mean concentration in the cross-section was also derived and evaluated in the same low-frequency limit. It was concluded that pulsatile flow maximises the axial dispersion compared with steady and purely oscillatory flows.