Complex Partial Differential Equations Research Articles

Solving partial differential equations using large-data models: a literature review

Mathematics lies at the heart of engineering science and is very important for capturing and modeling of diverse processes. These processes may be naturally-occurring or man-made. One important engineering problem in this regard is the modeling of advanced mathematical problems and their analysis. Partial differential equations (PDEs) are important and useful tools to this end. However, solving complex PDEs for advanced problems requires extensive computational resources and complex techniques. Neural networks provide a way to solve complex PDEs reliably. In this regard, large-data models are new generation of techniques, which have large dependency capturing capabilities. Hence, they can richly model and accurately solve such complex PDEs. Some common large-data models include Convolutional neural networks (CNNs) and their derivatives, transformers, etc. In this literature survey, the mathematical background is introduced. A gentle introduction to the area of solving PDEs using large-data models is given. Various state-of-the-art large-data models for solving PDEs are discussed. Also, the major issues and future scope of the area are identified. Through this literature survey, it is hoped that readers will gain an insight into the area of solving PDEs using large-data models and pursue future research in this interesting area.

Multistep Reduced Differential Transform Method in Solving Nonlinear Schrodinger Equations

This paper obtains semi-analytical solutions for the nonlinear Schrodinger equations (NLSEs) using the multistep reduced differential transform method (MsRDTM). The implemented method yields an analytical approximate solution over a longer time frame, in which the method applied is treated as an algorithm in a sequence of small sub-division of intervals of identical length compared to the traditional reduced differential transform method (RDTM). Excluding the need of perturbation, linearization, or discretization, this method offers the benefit and reliability of the multistep algorithm. The outcomes show that the MsRDTM generated highly accurate solutions of NLSEs than the RDTM. In addition, the results show that the suggested method is straightforward to use, saves a significant amount of computing work when solving NLSEs, and has potential for broad application in other complex partial differential equations (PDEs) in the fields of engineering and science. The accuracy of the method is shown through the tables and graphical illustrations provided.

Two-phase numerical simulation of thermal and solutal transport of zero mass flux conditions over a porous deformable disc: The extension of Jeffrey-Hamel model

In recent times, numerous assertions on the thermophysical properties of nanoliquids in various flow regimes, particularly the laminar flow regimes have been documented in the literature. With this focus, this theoretical investigation is to explore multiple solutions of the laminar two-dimensional Jeffrey fluid describing the Buongiorno nanofluid model and features of heat transport near a stagnation point across a movable radial disc. Also, the study seeks to explore the effects of zero mass flux and velocity of mass transpiration embedded within a Jeffrey fluid. With the help of the ansatz similarity transformations, the study simplifies complex partial differential equations into a set of ordinary differential equations, facilitating numerical dual solutions employing the bvp4c solver. The multiple solutions become apparent in a specific range of the shrinkable parameter. The results of the present study show that the suction parameter intensifies the phenomena of friction factor in the upper solution, whereas an annulment trend is observed in the lower solution. In addition, the shear stress and heat transfer rate decline with higher impacts of the Deborah number for the first solution while a monotonical behaviour is seen for the second solution. Besides, the absolute critical value decreases owing to the change value of the Deborah number, as a result, the separation of the boundary layer accelerates. Moreover, the heat transfer rate declined by about 0.29%, 0.37%, and 0.28% for the FBS and 0.30%, 0.32%, and 0.28% for the SBS with larger values of Nta, Nba and Sca, respectively. Lastly, the outcomes of the present examination with existing data are an excellent match for the limiting cases.

Antiphase and in-phase dynamics in laser chain models with delayed bidirectional couplings

The radiation dynamics of a closed chain of lasers with optoelectronic delayed coupling is analyzed. We consider bidirectional couplings of two designs: (i) diffusion type with feedback and (ii) bidirectional type without feedback. Assuming that the number of lasers is sufficiently large, we propose integro-differential models for spatially distributed variables with periodic boundary conditions. The critical value of the coupling coefficient is determined at which the stationary state of laser generation spontaneously becomes unstable due to the presence of a delay in the coupling lines. Bifurcations of asymptotically infinite dimension are described. For both types of coupling, we obtain the same two-dimensional complex partial differential equation of the Ginzburg–Landau type (with the difference only in the coefficients) for the slowly varying amplitude of the fundamental harmonic. Using the simplest homogeneous solution of such a quasi-normal form, the oscillations of laser radiation in chains, which can be anti-phase or in-phase depending on the time delay and the design of the couplings, are analytically described.

Results on solutions of several systems of the product type complex partial differential difference equations

Abstract This article is devoted to exploring the solutions of several systems of the first-order partial differential difference equations (PDDEs) with product type u ( z + c ) [ α 1 u ( z ) + β 1 u z 1 + γ 1 u z 2 + α 2 v ( z ) + β 2 v z 1 + γ 2 v z 2 ] = 1 , v ( z + c ) [ α 1 v ( z ) + β 1 v z 1 + γ 1 v z 2 + α 2 u ( z ) + β 2 u z 1 + γ 2 u z 2 ] = 1 , \left\{\begin{array}{l}u\left(z+c){[}{\alpha }_{1}u\left(z)+{\beta }_{1}{u}_{{z}_{1}}+{\gamma }_{1}{u}_{{z}_{2}}+{\alpha }_{2}v\left(z)+{\beta }_{2}{v}_{{z}_{1}}+{\gamma }_{2}{v}_{{z}_{2}}]=1,\\ v\left(z+c){[}{\alpha }_{1}v\left(z)+{\beta }_{1}{v}_{{z}_{1}}+{\gamma }_{1}{v}_{{z}_{2}}+{\alpha }_{2}u\left(z)+{\beta }_{2}{u}_{{z}_{1}}+{\gamma }_{2}{u}_{{z}_{2}}]=1,\end{array}\right. where c = ( c 1 , c 2 ) ∈ C 2 c=\left({c}_{1},{c}_{2})\in {{\mathbb{C}}}^{2} , α j , β j , γ j ∈ C , j = 1 , 2 {\alpha }_{j},{\beta }_{j},{\gamma }_{j}\in {\mathbb{C}},\hspace{0.33em}j=1,2 . Our theorems about the forms of the transcendental solutions for these systems of PDDEs are some improvements and generalization of the previous results given by Xu, Cao and Liu. Moreover, we give some examples to explain that the forms of solutions of our theorems are precise to some extent.

A fast general thermal simulation model based on Multi-Branch Physics-Informed deep operator neural network

Thermal simulation plays a crucial role in various fields, often involving complex partial differential equation (PDE) simulations for thermal optimization. To tackle this challenge, we have harnessed neural networks for thermal prediction, specifically employing deep neural networks as a universal solver for PDEs. This innovative approach has garnered significant attention in the scientific community. While Physics-Informed Neural Networks (PINNs) have been introduced for thermal prediction using deep neural networks, existing methods primarily focus on offering thermal simulations for predefined relevant parameters, such as heat sources, loads, boundaries, and initial conditions. However, any adjustments to these parameters typically require retraining or transfer learning, resulting in considerable additional work. To overcome this limitation, we integrated PINN methods with the DeepONet model, creating a novel model called PI-DeepONet for thermal simulation. This model takes both relevant parameters and coordinate points as simultaneous input functions, presenting a fresh computational perspective for thermal simulation. Based on the PaddlePaddle deep learning framework, our research demonstrates that after sufficient training, this model can reliably and rapidly predict parameter solutions. Importantly, it significantly surpasses traditional numerical solvers in terms of speed by several orders of magnitude, without requiring additional training. This groundbreaking research framework holds vast application potential and promises substantial advancements in the field of thermal simulation.

Virtual draw of microstructured optical fiber based on physics-informed neural networks.

The implementation of microstructured optical fibers (MOFs) with novel micro-structures and perfect performance is challenging due to the complex fabrication processes. Physics-informed neural networks (PINNs) offer what we believe to be a new approach to solving complex partial differential equations within the virtual fabrication model of MOFs. This study, for what appears to be the first time, integrates the complex partial differential equations and boundary conditions describing the fiber drawing process into the loss function of a neural network. To more accurately solve the free boundary of the fiber's inner and outer diameters, we additionally construct a neural network to describe the free boundary conditions. This model not only captures the evolution of the fiber's inner and outer diameters but also provides the velocity distribution and pressure distribution within the molten glass, thus laying the foundation for a quantitative analysis of capillary collapse. Furthermore, results indicate that the trends in the effects of temperature, feed speed, and draw speed on the fiber drawing process align with actual fabrication conditions, validating the feasibility of the model. The methodology proposed in this study offers what we believe to be a novel approach to simulating the fiber drawing process and holds promise for advancing the practical applications of MOFs.

Domain-decomposed Bayesian inversion based on local Karhunen-Loève expansions

In many Bayesian inverse problems the goal is to recover a spatially varying random field. Such problems are often computationally challenging especially when the forward model is governed by complex partial differential equations (PDEs). The challenge is particularly severe when the spatial domain is large and the unknown random field needs to be represented by a high-dimensional parameter. In this paper, we present a domain-decomposed method to attack the dimensionality issue and the method decomposes the spatial domain and the parameter domain simultaneously. On each subdomain, a local Karhunen-Loève (KL) expansion is constructed, and a local inversion problem is solved independently in a parallel manner, and more importantly, in a lower-dimensional space. After local posterior samples are generated through conducting Markov chain Monte Carlo (MCMC) simulations on subdomains, a novel projection procedure is developed to effectively reconstruct the global field. In addition, the domain decomposition interface conditions are dealt with an adaptive Gaussian process-based fitting strategy. Numerical examples are provided to demonstrate the performance of the proposed method.

Hemodynamic factors of spontaneous vertebral artery dissecting aneurysms assessed with numerical and deep learning algorithms: Role of blood pressure and asymmetry

Background and ObjectivesThe pathophysiology of spontaneous vertebral artery dissecting aneurysms (SVADA) is poorly understood. Our goal is to investigate the hemodynamic factors contributing to their formation using computational fluid dynamics (CFD) and deep learning algorithms. MethodsWe have developed software that can use patient imagery as input to recreate the vertebrobasilar arterial system, both with and without SVADA, which we used in a series of three patients. To obtain the kinematic blood flow data before and after the aneurysm forms, we utilized numerical methods to solve the complex Navier-Stokes partial differential equations. This was accomplished through the application of a finite volume solver (OpenFoam/Helyx OS). Additionally, we trained a neural ordinary differential equation (NODE) to learn and replicate the dynamical streamlines obtained from the computational fluid dynamics (CFD) simulations. ResultsIn all three cases, we observed that the equilibrium of blood pressure distributions across the VAs, at a specific vertical level, accurately predicted the future SVADA location. In the two cases where there was a dominant VA, the dissection occurred on the dominant artery where blood pressure was lower compared to the contralateral side. The SVADA sac was characterized by reduced wall shear stress (WSS) and decreased velocity magnitude related to increased turbulence. The presence of a high WSS gradient at the boundary of the SVADA may explain its extension. Streamlines generated by CFD were learned with a neural ordinary differential equation (NODE) capable of capturing the system’s dynamics to output meaningful predictions of the flow vector field upon aneurysm formation. ConclusionIn our series, asymmetry in the vertebrobasilar blood pressure distributions at and proximal to the site of the future SVADA accurately predicted its location in all patients. Deep learning algorithms can be trained to model blood flow patterns within biological systems, offering an alternative to the computationally intensive CFD. This technology has the potential to find practical applications in clinical settings.

A DOMAIN-DECOMPOSED VAE METHOD FOR BAYESIAN INVERSE PROBLEMS

Bayesian inverse problems are often computationally challenging when the forward model is governed by complex partial differential equations (PDEs). This is typically caused by expensive forward model evaluations and highdimensional parameterization of priors. This paper proposes a domain-decomposed variational autoencoder Markov chain Monte Carlo (DD-VAE-MCMC) method to tackle these challenges simultaneously. Through partitioning the global physical domain into small subdomains, the proposed method first constructs local deterministic generative models based on local historical data, which provide efficient local prior representations. Gaussian process models with active learning address the domain decomposition interface conditions. Then inversions are conducted on each subdomain independently in parallel and in low-dimensional latent parameter spaces. The local inference solutions are postprocessed through the Poisson image blending procedure to result in an efficient global inference result. Numerical examples are provided to demonstrate the performance of the proposed method.

The inhomogeneous complex partial differential equations for bi-polyanalytic functions

<abstract><p>In this paper, we study a Riemann-Hilbert problem related to inhomogeneous complex partial differential operators of higher order on the unit disk. Applying the Cauchy-Pompeiu formula, we find out the solvable conditions and obtain the representation of the solutions. Then, we investigate the boundary value problems for bi-polyanalytic functions with the Dirichlet and Riemann-Hilbert boundary conditions, obtain the specific solution and the solvable conditions, and extend the conclusion to the corresponding higher-order problems. Therefore, we obtain the solution to the half-Neumann problem of higher order for bi-polyanalytic functions.</p></abstract>

A general unsteady Fourier solution for orthotropic heat transfer in 2D functionally graded cylinders

In the present research, an analytical solution is presented for unsteady heat transfer in a two‐dimensional finite cylinder made of functionally graded Materials (FGM). The governing equation, as an unsteady partial differential equation (PDE), is solved with general boundary conditions (BCs). The thermal conductivity coefficients are considered as a power function of the radius, both in the longitudinal and radial directions of the FGM cylinder. The presence of non‐constant coefficients and general BCs will confront us with the challenge of solving an unsteady problem with high complexity. A combination of Laplace transform, Fourier transform, and meromorphic functions are utilized to deal with this complex PDE. The Laplace transform was applied to transform to the frequency domain, the Sturm–Liouville equation was used to extract the orthogonal basis Fourier series, and the meromorphic function was deployed to transform back from the frequency domain to the time domain. Results were well verified against previous research works. Results showed that the proposed analytical solution could well predict the temperature distribution. The current analytical solution is useful to better understand unsteady heat transfer in FGMs.

Entropy analysis of electromagnetic trihybrid nanofluid flow with temperature-dependent viscosity in a Darcy-Forchheimer porous medium over a stretching sheet under convective conditions

This study introduces a novel theoretical model known as the trihybrid nanofluid (THNF), which demonstrates remarkable thermal conductivity properties to enhance heat transfer in liquids. The base liquid (H2O) was employed to immerse nanoparticles of Cu, Al2O3, and TiO2, thereby resulting in the creation of a THNF (Cu + Al2O3 + TiO2/H2O). This research focuses on the entropy generation of electromagnetic THNF flow in a Darcy-Forchheimer porous medium over a stretching sheet under convective conditions. The analysis considers the intricate interplay between various factors, including convection, radiation, heat source, chemical processes, and cross-diffusion effects. Complex partial differential equations for flow, energy, and concentration are transformed into nonlinear ordinary differential equations using similarity transformations. The Runge–Kutta–Fehlberg method with a shooting mechanism is used to solve these ODEs. The comprehensive graphical analysis visually exemplifies the impact of significant parameters on the profiles of velocity, temperature, and concentration, in conjunction with skin friction, Nusselt, and Sherwood numbers. The flow velocity decreased by 8.89 % in NFs with higher viscosity and Darcy-Forchheimer porous medium values, but it accelerated by 13.64 % in THNF with more convection. Electric fields boost THNF velocity and temperature by 20.94 % and 14.27 % respectively, while stronger magnetic fields reduce NF velocity by 15.34 % and elevate THNF temperature by 7.52 %. THNFs show the largest temperature increase of 18.67 % due to radiation, while it decreases by more than 7.51 % for NFs with an increasing Prandtl number. NP concentration drops by 10.19 % when Schmidt and chemical reactions increase. Elevating electric and magnetic fields can reduce heat transfer by 4.78 % and entropy generation by at least 6.45 % in NFs, while increasing the Bejan number by up to 18.83 % in THNFs. Radiation and diffusion increase the entropy generation by 6.71 % and decrease the Bejan number by 7.98 %.

Enhancing physics informed neural networks for solving Navier–Stokes equations

SummaryFluid mechanics is a critical field in both engineering and science. Understanding the behavior of fluids requires solving the Navier–Stokes equation (NSE). However, the NSE is a complex partial differential equation that can be challenging to solve, and classical numerical methods can be computationally expensive. In this paper, we propose enhancing physics‐informed neural networks (PINNs) by modifying the residual loss functions and incorporating new computational deep learning techniques. We present two enhanced models for solving the NSE. The first model involves developing the classical PINN for solving the NSE, based on a stream function approach to the velocity components. We have added the pressure training loss function to this model and integrated the new computational training techniques. Furthermore, we propose a second, more flexible model that directly approximates the solution of the NSE without making any assumptions. This model significantly reduces the training duration while maintaining high accuracy. Moreover, we have successfully applied this model to solve the three‐dimensional NSE. The results demonstrate the effectiveness of our approaches, offering several advantages, including high trainability, flexibility, and efficiency.

Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2

Abstract Our purpose in this article is to describe the solutions of several product-type nonlinear partial differential equations (PDEs) ( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = 1 , \left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})=1, and ( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = e g , \left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})={e}^{g}, where g ( z ) g\left(z) is a nonconstant polynomial and a j , b j {a}_{j},{b}_{j} , and c j ( j = 1 , 2 ) {c}_{j}\left(j=1,2) are constants in C {\mathbb{C}} . The finite-order transcendental entire solution u u of the first equation is of the following forms: u ( z 1 , z 2 ) = ± 1 a 1 a 2 + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] , u\left({z}_{1},{z}_{2})=\pm \frac{1}{\sqrt{{a}_{1}{a}_{2}}}+{\eta }_{0}{e}^{\tfrac{1}{D}{[}\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]}, or u ( z 1 , z 2 ) = 1 2 a 1 e Q ( z 1 , z 2 ) + 1 2 a 2 e − Q ( z 1 , z 2 ) + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] , u\left({z}_{1},{z}_{2})=\frac{1}{2{a}_{1}}{e}^{Q\left({z}_{1},{z}_{2})}+\frac{1}{2{a}_{2}}{e}^{-Q\left({z}_{1},{z}_{2})}+{\eta }_{0}{e}^{\tfrac{1}{D}{[}\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]}, where D = b 1 c 2 − b 2 c 1 D={b}_{1}{c}_{2}-{b}_{2}{c}_{1} , η 0 ∈ C − { 0 } {\eta }_{0}\in {\mathbb{C}}-\left\{0\right\} , and Q ( z 1 , z 2 ) = − 1 D [ ( a 1 c 2 + a 2 c 1 ) z 1 − ( a 1 b 2 + a 2 b 1 ) z 2 ] + η 1 , η 1 ∈ C . Q\left({z}_{1},{z}_{2})=-\frac{1}{D}\left[\left({a}_{1}{c}_{2}+{a}_{2}{c}_{1}){z}_{1}-\left({a}_{1}{b}_{2}+{a}_{2}{b}_{1}){z}_{2}]+{\eta }_{1},\hspace{1em}{\eta }_{1}\in {\mathbb{C}}. The description of the forms of the solutions for these PDEs demonstrates that our results are some improvements of the previous results given by Liu, Cao, and Xu [L. Xu and T. B. Cao, Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math. 15 (2018), 227], and [K. Liu and T. B. Cao, Entire solutions of Fermat type difference differential equations, Electron. J. Diff. Equ. 2013 (2013), No. 59, 1–10.]. Meantime, we list some examples to explain that the forms of solutions of our theorems are precise to some extent.

Physics informed WNO

Deep neural operators are recognized as an effective tool for learning solution operators of complex partial differential equations (PDEs). As compared to laborious analytical and computational tools, a single neural operator can predict solutions of PDEs for varying initial or boundary conditions and different inputs without repeated independent runs from scratch. A recently proposed Wavelet Neural Operator (WNO) is one such operator that harnesses the advantage of time–frequency localization of wavelets to capture the manifolds in the spatial domain effectively. While WNO has proven to be a promising method for operator learning, the data-hungry nature of the framework is a major shortcoming. Relying completely on conventional solvers for data generation and subsequently training operators with generated data leads to a time-consuming and challenging implementation of the operators in practical applications. In this work, we propose a physics-informed WNO for learning the solution operators of families of parametric PDEs without labeled training data. The efficacy of the framework is validated and illustrated with four nonlinear spatiotemporal systems relevant to various fields of engineering and science.

Empowering nuclear reactor analysis: OpenNode for accurate 3D core simulation through nodal expansion neutron diffusion equations solving

OpenNode is a sophisticated reactor core simulation tool, meticulously designed for educational purposes. Its main function is to solve the complex partial differential equation governing multi-group neutron diffusion in a Cartesian geometry. OpenNode relies on the second- and fourth-order Nodal Expansion Method (NEM) to provide accurate solutions to three-dimensional steady-state problems, encompassing both the direct and adjoint modes. At the heart of this method, the neutron balance equation serves as the fundamental framework for calculating the nodal neutron flux and the effective multiplication factor, enabling the complex evaluation of neutron flux distributions within individual nodes.The NEM methodology, carefully programmed in Fortran-90 in OpenNode, meticulously decomposes the reactor core into nodes of reasonable size. This approach balances computational efficiency and accuracy, achieving outstanding precision even when using a reduced number of computational meshes. To further increase the number of equations while maintaining the balance between unknowns, OpenNode judiciously incorporates two essential approximations: the Weighted Residual Method (WRM) and the Quadratic Transverse Leakage (QTL) approximation.OpenNode transcends simple computational prowess by adopting an intuitive Graphical User Interface (GUI), meticulously developed by combining Python 3, Qt Designer, and Blender software. This Python-powered graphical interface offers users a seamless, user-friendly experience, facilitating tasks such as data entry, execution of complex simulations, calculation of results, data visualization, 3D design of nuclear core geometry, and in-depth analysis of results. One of OpenNode’s key features is its innate flexibility, enabling it to solve problems in both 2D and 3D Cartesian geometries. This adaptability makes it a versatile tool suitable for a wide range of research and power reactor applications, offering users the ability to model and design reactors with unrivaled ease.To substantiate OpenNode’s reliability, comprehensive criticality calculations are carried out in various scenarios, including single-cell verification, BIBLIS, KOEBERG cores in two dimensions (2D), and IAEA cores in three dimensions (3D). The results clearly demonstrate OpenNode’s robustness and reliability in multidimensional criticality analysis. In particular, OpenNode adheres to an open-source philosophy, making it readily available for download. This commitment to open access fosters collaboration and paves the way for wider adoption within the education and research communities, nurturing a thriving ecosystem of shared knowledge and exploration.

Stress state and waves in the lithospheric plate simulation: A 3rd generation AI architecture

Natural disasters present ongoing risks to human life and the global economy, with climate change and environmental factors exacerbating these threats. This article introduces an innovative approach to earthquake prediction and modeling, utilizing a combination of modern mathematical techniques, data-driven methodologies, and artificial intelligence (AI). By integrating the finite element method and wavelet collocation method, our approach enables the solution of complex partial differential equations involving fractional derivatives. This integration provides a more accurate prediction of earthquake behavior by leveraging AI analysis of the resulting numerical solutions.Our advanced modeling technique enables three-dimensional earthquake modeling, surpassing the limitations of traditional methods and offering unprecedented insights. The comprehensive AI approach shows great potential for improving our understanding of earthquake behavior, facilitating the design of earthquake-resistant structures, and effectively mitigating the devastating impacts of earthquakes.This research significantly contributes to the field of disaster risk reduction and offers practical applications for both researchers and practitioners. By combining mathematical rigor, data-driven insights, and the power of AI, our next-generation approach opens new avenues for enhancing our preparedness and response to seismic events.

A Solution of the Complex Fuzzy Heat Equation in Terms of Complex Dirichlet Conditions Using a Modified Crank–Nicolson Method

Complex fuzzy sets (CFSs) have recently emerged as a potent tool for expanding the scope of fuzzy sets to encompass wider ranges within the unit disk in the complex plane. This study explores complex fuzzy numbers and introduces their application for the first time in the literature to address a complex fuzzy partial differential equation that involves a complex fuzzy heat equation under Hukuhara differentiability. The researchers utilize an implicit finite difference scheme, namely the Crank–Nicolson method, to tackle complex fuzzy heat equations. The problem’s fuzziness arises from the coefficients in both amplitude and phase terms, as well as in the initial and boundary conditions, with the Convex normalized triangular fuzzy numbers extended to the unit disk in the complex plane. The researchers take advantage of the properties and benefits of CFS theory in the proposed numerical methods and subsequently provide a new proof of the stability under CFS theory. A numerical example is presented to demonstrate the proposed approach’s reliability and feasibility, with the results showing good agreement with the exact solution and relevant theoretical aspects.

How Can (or Why Should) Process Engineering Aid the Screening and Discovery of Solid Sorbents for CO2 Capture?

ConspectusAdsorption using solid sorbents is emerging as a serious contender to amine-based liquid absorption for postcombustion CO2 capture. In the last 20+ years, significant efforts have been invested in developing adsorption processes for CO2 capture. In particular, significant efforts have been invested in developing new adsorbents for this application. These efforts have led to the generation of hundreds of thousands of (hypothetical and real) adsorbents, e.g., zeolites and metal-organic frameworks (MOFs). Identifying the right adsorbent for CO2 capture remains a challenging task. Most studies are focused on identifying adsorbents based on certain adsorption metrics. Recent studies have demonstrated that the performance of an adsorbent is intimately linked to the process in which it is deployed. Any meaningful screening should thus consider the complexity of the process. However, simulation and optimization of adsorption processes are computationally intensive, as they constitute the simultaneous propagation of heat and mass transfer fronts; the process is cyclic, and there are no straightforward design tools, thereby making large-scale process-informed screening of sorbents prohibitive.This Account discusses four papers that develop computational methods to incorporate process-based evaluation for both bottom-up (chemistry to engineering) screening problems and top-down (engineering to chemistry) inverse problems. We discuss the development of the machine-assisted adsorption process learning and emulation (MAPLE) framework, a surrogate model based on deep artificial neural networks (ANNs) that can predict process-level performance by considering both process and material inputs. The framework, which has been experimentally validated, allows for reliable, process-informed screening of large adsorbent databases. We then discuss how process engineering tools can be used beyond adsorbent screening, i.e., to estimate the practically achievable performance and cost limits of pressure vacuum swing adsorption (PVSA) processes should the ideal bespoke adsorbent be made. These studies show what conditions stand-alone PVSA processes are attractive and when they should not be considered. Finally, recent developments in physics-informed neural networks (PINNS) enable the rapid solution of complex partial differential equations, providing tools to potentially identify optimal cycle configurations. Ultimately, we provide areas where further developments are required and emphasize the need for strong collaborations between chemists and chemical engineers to move rapidly from discovery to field trials, as we do not have much time to fulfill commitments to net-zero targets.