Solution Of Partial Differential Equations Research Articles

A Novel Hybrid Deep Learning Method for Predicting the Flow Fields of Biomimetic Flapping Wings.

The physics governing the fluid dynamics of bio-inspired flapping wings is effectively characterized by partial differential equations (PDEs). Nevertheless, the process of discretizing these equations at spatiotemporal scales is notably time consuming and resource intensive. Traditional PDE-based computations are constrained in their applicability, which is mainly due to the presence of numerous shape parameters and intricate flow patterns associated with bionic flapping wings. Consequently, there is a significant demand for a rapid and accurate solution to nonlinear PDEs, to facilitate the analysis of bionic flapping structures. Deep learning, especially physics-informed deep learning (PINN), offers an alternative due to its great nonlinear curve-fitting capability. In the present work, a hybrid coarse-data-driven physics-informed neural network model (HCDD-PINN) is proposed to improve the accuracy and reliability of predicting the time evolution of nonlinear PDEs solutions, by using an order-of-magnitude-coarser grid than traditional computational fluid dynamics (CFDs) require as internal training data. The architecture is devised to enforce the initial and boundary conditions, and incorporate the governing equations and the low-resolution spatiotemporal internal data into the loss function of the neural network, to drive the training. Compared to the original PINN with no internal data, the training and predicting dynamics of HCDD-PINN with different resolutions of coarse internal data are analyzed on the problem relevant to the two-dimensional unsteady flapping wing, which involves unsteady flow features and moving boundaries. Additionally, a hyper-parametrical study is conducted to obtain an optimal model for the problem under consideration, which is then utilized for investigating the effects of the snapshot and fraction of the coarse internal data on the HCDD-PINN's performances. The results show that the proposed framework has a sufficient stability and accuracy for solving the considered biomimetic flapping-wing problem, and its great potential means that it can be considered as an alternative to accelerate or replace traditional CFD solvers in the future. The interested variables of the flow field at any instant can be rapidly obtained by the trained HCDD-PINN model, which is superior to the traditional CFD method that usually needs to be re-run. For the three-dimensional and optimization problems of flapping wings, the advantages of the proposed method are supposedly even more apparent.

TCAS-PINN: Physics-informed neural networks with a novel temporal causality-based adaptive sampling method

Physics-informed neural networks (PINNs) have become an attractive machine learning framework for obtaining solutions to partial differential equations (PDEs). PINNs embed initial, boundary, and PDE constraints into the loss function. The performance of PINNs is generally affected by both training and sampling. Specifically, training methods focus on how to overcome the training difficulties caused by the special PDE residual loss of PINNs, and sampling methods are concerned with the location and distribution of the sampling points upon which evaluations of PDE residual loss are accomplished. However, a common problem among these original PINNs is that they omit special temporal information utilization during the training or sampling stages when dealing with an important PDE category, namely, time-dependent PDEs, where temporal information plays a key role in the algorithms used. There is one method, called Causal PINN, that considers temporal causality at the training level but not special temporal utilization at the sampling level. Incorporating temporal knowledge into sampling remains to be studied. To fill this gap, we propose a novel temporal causality-based adaptive sampling method that dynamically determines the sampling ratio according to both PDE residual and temporal causality. By designing a sampling ratio determined by both residual loss and temporal causality to control the number and location of sampled points in each temporal sub-domain, we provide a practical solution by incorporating temporal information into sampling. Numerical experiments of several nonlinear time-dependent PDEs, including the Cahn–Hilliard, Korteweg–de Vries, Allen–Cahn and wave equations, show that our proposed sampling method can improve the performance. We demonstrate that using such a relatively simple sampling method can improve prediction performance by up to two orders of magnitude compared with the results from other methods, especially when points are limited.

On The Double Integral Transform (Complex EE Transform) and Their Properties and Applications

Due to the importance of solutions of partial differential equations, linear, nonlinear, homogeneous, and non-homogeneous, in important life applications, including engineering applications, physics and astronomy, medical sciences, and life technology, and their importance in solutions to heat transfer equations, wave, Laplace equation, telegraph, etc. In this paper, a new double integral transform has been proposed. In this work, we have introduced a new double transform ( Double Complex EE Transform ). In addition, we presented the convolution theorem and proved the properties of the proposed transform, which has an effective and useful role in dealing with the solution of two-dimensional partial differential equations. Moreover, two examples of important mathematical equations are solved to illustrate method. This double integral transformation has a complex kernel.

Compressive Fourier collocation methods for high-dimensional diffusion equations with periodic boundary conditions

Abstract High-dimensional partial differential equations (PDEs) are a popular mathematical modelling tool, with applications ranging from finance to computational chemistry. However, standard numerical techniques for solving these PDEs are typically affected by the curse of dimensionality. In this work, we tackle this challenge while focusing on stationary diffusion equations defined over a high-dimensional domain with periodic boundary conditions. Inspired by recent progress in sparse function approximation in high dimensions, we propose a new method called compressive Fourier collocation. Combining ideas from compressive sensing and spectral collocation, our method replaces the use of structured collocation grids with Monte Carlo sampling and employs sparse recovery techniques, such as orthogonal matching pursuit and $\ell ^1$ minimization, to approximate the Fourier coefficients of the PDE solution. We conduct a rigorous theoretical analysis showing that the approximation error of the proposed method is comparable with the best $s$-term approximation (with respect to the Fourier basis) to the solution. Using the recently introduced framework of random sampling in bounded Riesz systems, our analysis shows that the compressive Fourier collocation method mitigates the curse of dimensionality with respect to the number of collocation points under sufficient conditions on the regularity of the diffusion coefficient. We also present numerical experiments that illustrate the accuracy and stability of the method for the approximation of sparse and compressible solutions.

Optimal Incentives in a Limit Order Book: A SPDE Control Approach

With the fragmentation of electronic markets, exchanges are now competing in order to attract trading activity on their platform. Consequently, they developed several regulatory tools to control liquidity provision/consumption on their liquidity pool. In this paper, we study the problem of an exchange using incentives in order to increase market liquidity. We model the limit order book as the solution of a stochastic partial differential equation (SPDE) as in [R. Cont and M. S. Müller, 2021, A stochastic partial differential equation model for limit order book dynamics, SIAM Journal on Financial Mathematics 12(2), 744–787]. The incentives proposed to the market participants are functions of the time and the distance of their limit order to the mid-price. We formulate the control problem of the exchange who wishes to modify the shape of the order book by increasing the volume at specific limits. Due to the particular nature of the SPDE control problem, we are able to characterize the solution with a classic Feynman–Kac representation theorem. Moreover, when studying the asymptotic behavior of the solution, a specific penalty function enables the exchange to obtain closed-form incentives at each limit of the order book. We study numerically the form of the incentives and their impact on the shape of the order book, and analyze the sensitivity of the incentives to the market parameters.

Efficient physics-informed neural networks using hash encoding

Physics-informed neural networks (PINNs) have attracted much attention in scientific computing as their functional representation of partial differential equation (PDE) solutions offers flexibility and accuracy features. However, their training cost has limited their practical use as a real alternative to classic numerical methods. Thus, we propose to incorporate multi-resolution hash encoding into PINNs to improve the training efficiency, as such encoding offers a locally-aware (at multi-resolution) coordinate input to the neural network. Borrowed from the neural representation field community (NeRF), we investigate the robustness of calculating the derivatives of such hash-encoded neural networks with respect to the input coordinates, which is often needed by the PINN loss terms. We propose to replace the automatic differentiation with finite-difference calculations of the derivatives to address the discontinuous nature of such derivatives. We also share the appropriate ranges for the hash encoding hyperparameters to obtain robust derivatives. We test the proposed method on several benchmark problems, and the proposed method admits about a 10-fold improvement in efficiency over the vanilla PINN implementation.

Modelling count data with partial differential equation models in biology

Partial differential equation (PDE) models are often used to study biological phenomena involving movement-birth–death processes, including ecological population dynamics and the invasion of populations of biological cells. Count data, by definition, is non-negative, and count data relating to biological populations is often bounded above by some carrying capacity that arises through biological competition for space or nutrients. Parameter estimation, parameter identifiability, and making model predictions usually involves working with a measurement error model that explicitly relating experimental measurements with the solution of a mathematical model. In many biological applications, a typical approach is to assume the data are normally distributed about the solution of the mathematical model. Despite the widespread use of the standard additive Gaussian measurement error model, the assumptions inherent in this approach are rarely explicitly considered or compared with other options. Here, we interpret scratch assay data, involving migration, proliferation and delays in a population of cancer cells using a reaction–diffusion PDE model. We consider relating experimental measurements to the PDE solution using a standard additive Gaussian measurement error model alongside a comparison to a more biologically realistic binomial measurement error model. While estimates of model parameters are relatively insensitive to the choice of measurement error model, model predictions for data realisations are very sensitive. The standard additive Gaussian measurement error model leads to biologically inconsistent predictions, such as negative counts and counts that exceed the carrying capacity across a relatively large spatial region within the experiment. Furthermore, the standard additive Gaussian measurement error model requires estimating an additional parameter compared to the binomial measurement error model. In contrast, the binomial measurement error model leads to biologically plausible predictions and is simpler to implement. We provide open source Julia software on GitHub to replicate all calculations in this work, and we explain how to generalise our approach to deal with coupled PDE models with several dependent variables through a multinomial measurement error model, as well as pointing out other potential generalisations by linking our work with established practices in the field of generalised linear models.

Determining a parabolic system by boundary observation of its non-negative solutions with biological applications

In this paper, we consider the inverse problem of determining some coefficients within a coupled nonlinear parabolic system, through boundary observation of its non-negative solutions. In the physical setup, the non-negative solutions represent certain probability densities in different contexts. We innovate the successive linearisation method by further developing a high-order variation scheme which can both ensure the positivity of the solutions and effectively tackle the nonlinear inverse problem. This enables us to establish several novel unique identifiability results for the inverse problem in a rather general setup. For a theoretical perspective, our study addresses an important topic in partial differential equation (PDE) analysis on how to characterise the function spaces generated by the products of non-positive solutions of parabolic PDEs. As a typical and practically interesting application, we apply our general results to inverse problems for ecological population models, where the positive solutions signify the population densities.

Usage of the Fuzzy Adomian Decomposition Method for Solving Some Fuzzy Fractional Partial Differential Equations

In this study, we examine the numerical solutions of nonlinear fuzzy fractional partial differential equations under the Caputo derivative utilizing the technique of fuzzy Adomian decomposition. This technique is used as an alternative method for obtaining approximate fuzzy solutions to various types of fractional differential equations and also investigated some new existence and uniqueness results of fuzzy solutions. Some examples are given to support the effectiveness of the proposed technique. We present the numerical results in graphical form for different values of fractional order and uncertainty γ∈0,1.

Physics-informed kernel function neural networks for solving partial differential equations

This paper proposes an improved version of physics-informed neural networks (PINNs), the physics-informed kernel function neural networks (PIKFNNs), to solve various linear and some specific nonlinear partial differential equations (PDEs). It can also be considered as a novel radial basis function neural network (RBFNN). In the proposed PIKFNNs, it employs one-hidden-layer shallow neural network with the physics-informed kernel functions (PIKFs) as the customized activation functions. The PIKFs fully or partially contain PDE information, which can be chosen as fundamental solutions, green's functions, T-complete functions, harmonic functions, radial Trefftz functions, probability density functions and even the solutions of some linear simplified PDEs and so on. The main difference between the PINNs and the proposed PIKFNNs is that the PINNs add PDE constraints to the loss function, and the proposed PIKFNNs embed PDE information into the activation functions of the neural network. The feasibility and accuracy of the proposed PIKFNNs are validated by some benchmark examples.

Approximation of Functions Using Fourier Series and Its Application to the Solution of Partial Differential Equations

Approximation of Functions Using Fourier Series and Its Application to the Solution of Partial Differential Equations

A NEW PERSPECTIVE ON THE NONLINEAR DATE–JIMBO–KASHIWARA–MIWA EQUATION IN FRACTAL MEDIA

In this paper, we first created a fractal Date–Jimbo–Kashiwara–Miwa (FDJKM) long ripple wave model in a non-smooth boundary or microgravity space recorded. Using fractal semi-inverse skill (FSIS) and fractal traveling wave transformation (FTWT), the fractal variational principle (FVP) was derived, and the strong minimum necessary circumstance was attested with the He Wierstrass function. We have discovered two distinct solitary wave solutions, the square form of the hyperbolic secant function and the hyperbolic secant function form. Then, soliton solutions are cultivated through FVP and the minimum steady state condition. Finally, the influences of non-smooth boundaries on solitons were tackled, and the properties of the solution were demonstrated through three-dimensional contour lines. Fractal dimension can impact waveforms, but cannot affect their vertex values. The presentation of soliton solutions (SWS) using techniques is not only laudable but also noteworthy. The technique employed can also be used to investigate solitary wave solutions of other local fractional calculus partial differential equations.

Lie symmetry analysis and exact solutions for conformable time fractional partial differential equations with variable coefficients

Lie symmetry analysis and exact solutions for conformable time fractional partial differential equations with variable coefficients

Solving first order fuzzy partial differential equations using fuzzy Aboodh transform

In this article, we extend the concept of the fuzzy Aboodh transform to the solution of fuzzy partial differential equations using Hukuhara differentiability. In order to create the solution to first-order fuzzy partial differential equations, we first provide the fuzzy Aboodh transform and then present the basic definitions and theorems for the fuzzy Aboodh transform of fuzzy partial derivatives. Finally, the method is illustrated with an example to show the ability of the proposed method.

On Developing He Method with Mohand Transform to Find Numerical and Exact Solutions of Some Neutrosophic Partial Differential Equations

In this paper, we propose a novel combination of the (He) method with (Mohand transform), and this combination is summarized by representing the nonlinear part (or the residue of the operator) with (He) polynomials after applying the Mohand transform. To prove the accuracy of the proposed method, we applied it to several neutrosophic partial differential equations such as neutrosophic version of Helmholtz equation, neutrosophic version of non-linear oscillator, neutrosophic version of Burger's equation, and the neutrosophic version of telegraph equation. The accuracy and effectiveness of the application of the proposed method was verified by comparing the results obtained with other methods using the Maple software package.

On the Usage of Orthogonal Polynomials with Picard Iteration Method to Find Numerical Solutions of Neutrosophic Non-Linear Ordinary and Partial Differential Equations

This research aims to modify the Picard iteration method by hybridizing it with some orthogonal polynomials and then applying the hybrid method in solving neutrosophic nonlinear elementary value problems. This method is based on modifying the Picard iteration method by approximating the right-hand side of the neutrosophic differential equation of the studied problem either by Legendre polynomials or by Chebyshev polynomials of the first kind to obtain two different hybrids of the Picard iteration method. Also, we apply this modification to neutrosophic elementary value problems represented by neutrosophic nonlinear and right-handed nonlinear differential equations to demonstrate the reliability and efficiency of the proposed modified method. For this goal, we prove how effective this method is, we calculate the neutrosophic absolute error of approximate solutions resulting from the application of the proposed modification of the Picard iteration method and with the exact solution.

On some extrapolation in generalized grand Morrey spaces with applications to PDEs

<abstract><p>Rubio de Francia's extrapolation in generalized grand Morrey spaces is derived. This result is applied to the investigation of the regularity of solutions for the second order partial differential equations with discontinuous coefficients in the framework of generalized grand Morrey spaces under the Muckenhoupt condition on weights. Density properties for these spaces are also investigated.</p></abstract>

EXACT TRAVELING WAVE SOLUTION OF GENERALIZED (4+1)-DIMENSIONAL LOCAL FRACTIONAL FOKAS EQUATION

In this paper, within the scope of the local fractional derivative theory, the (4+1)-dimensional local fractional Fokas equation is researched. The study of exact solutions of high-dimensional nonlinear partial differential equations plays an important role in understanding complex physical phenomena in reality. In this paper, the exact traveling wave solution of generalized functions is analyzed defined on Cantor sets in high-dimensional integrable systems. The results of non-differentiable solutions in different cases are numerically simulated when the fractal dimension is equal to [Formula: see text] = ln 2/ln 3. The results show that the exact solution of the local fractional Fokas equation represents the fractal waves on the shallow water surface. Through numerical simulation, we find that the exact solution of the local fractional Fokas equation can describe the fractal waves and waves characteristics of shallow water surface. It also shows that the study of traveling wave solutions of nonlinear local fractional equations has important significance in mathematical physics.

Existence of solutions of fractal fractional partial differential equations through different contractions

<abstract><p>In the past, the existence and uniqueness of the solutions of fractional differential equations have been investigated by many researchers theoretically in various approaches in the literature. In this paper, there is no discussion of the existence of solutions for the nonlinear differential equations with fractal fractional operators. The objective of this work is to present novel contraction approaches, notably the $ \varpropto $-$ \psi $-contraction $ \varpropto $-type of the $ \tilde{\texttt{F}} $-contraction, within the context of $ \hat{F} $-metric and orbital metric spaces. The aim of this study is to illustrate certain fixed point theorems that offer a new and direct approach to establish the existence and uniqueness of the solution to the general partial differential equations by employing the fractal fractional operators.</p></abstract>

A DIMENSION-ADAPTIVE COMBINATION TECHNIQUE FOR UNCERTAINTY QUANTIFICATION

We present an adaptive algorithm for the computation of quantities of interest involving the solution of a stochastic elliptic partial differential equation, where the diffusion coefficient is parametrized by means of a Karhunen-Loève expansion. The approximation of the equivalent parametric problem requires a restriction of the countably infinite-dimensional parameter space to a finite-dimensional parameter set, a spatial discretization, and an approximation in the parametric variables. We consider a sparse grid approach between these approximation directions in order to reduce the computational effort and propose a dimension-adaptive combination technique. In addition, a sparse grid quadrature for the high-dimensional parametric approximation is employed and simultaneously balanced with the spatial and stochastic approximation. Our adaptive algorithm constructs a sparse grid approximation based on the benefit-cost ratio such that the regularity and thus the decay of the Karhunen-Loève coefficients is not required beforehand. The decay is detected and exploited as the algorithm adjusts to the anisotropy in the parametric variables. We include numerical examples for the Darcy problem with a lognormal permeability field, which illustrate a good performance of the algorithm. For sufficiently smooth random fields, we essentially recover the spatial order of convergence as asymptotic convergence rate with respect to the computational cost.