Family Of Partial Differential Equations Research Articles

Learning Traveling Solitary Waves Using Separable Gaussian Neural Networks.

In this paper, we apply a machine-learning approach to learn traveling solitary waves across various physical systems that are described by families of partial differential equations (PDEs). Our approach integrates a novel interpretable neural network (NN) architecture, called Separable Gaussian Neural Networks (SGNN) into the framework of Physics-Informed Neural Networks (PINNs). Unlike the traditional PINNs that treat spatial and temporal data as independent inputs, the present method leverages wave characteristics to transform data into the so-called co-traveling wave frame. This reformulation effectively addresses the issue of propagation failure in PINNs when applied to large computational domains. Here, the SGNN architecture demonstrates robust approximation capabilities for single-peakon, multi-peakon, and stationary solutions (known as "leftons") within the (1+1)-dimensional, b-family of PDEs. In addition, we expand our investigations, and explore not only peakon solutions in the ab-family but also compacton solutions in (2+1)-dimensional, Rosenau-Hyman family of PDEs. A comparative analysis with multi-layer perceptron (MLP) reveals that SGNN achieves comparable accuracy with fewer than a tenth of the neurons, underscoring its efficiency and potential for broader application in solving complex nonlinear PDEs.

Waveformer for modeling dynamical systems

Neural operators have gained recognition as potent tools for learning solutions of a family of partial differential equations. The state-of-the-art neural operators excel at approximating the functional relationship between input functions and the solution space, potentially reducing computational costs and enabling real-time applications. However, they often fall short when tackling time-dependent problems, particularly in delivering accurate long-term predictions. In this work, we propose “waveformer”, a novel operator learning approach for learning solutions of dynamical systems. The proposed waveformer exploits wavelet transform to capture the spatial multi-scale behavior of the solution field and transformers for capturing the long horizon dynamics. We present four numerical examples involving Burgers’s equation, KS-equation, Allen–Cahn equation, and Navier–Stokes equation to illustrate the efficacy of the proposed approach. Results obtained indicate the capability of the proposed waveformer in learning the solution operator and show that the proposed Waveformer can learn the solution operator with high accuracy, outperforming existing state-of-the-art operator learning algorithms by up to an order, with its advantage particularly visible in the extrapolation region.

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.

Bifurcations and Early-Warning Signs for SPDEs with Spatial Heterogeneity

AbstractBistability is a key property of many systems arising in the nonlinear sciences. For example, it appears in many partial differential equations (PDEs). For scalar bistable reaction-diffusions PDEs, the bistable case even has taken on different names within communities such as Allee, Allen-Cahn, Chafee-Infante, Nagumo, Ginzburg-Landau,$$\Phi ^4$$Φ4, Schlögl, Stommel, just to name a few structurally similar bistable model names. One key mechanism, how bistability arises under parameter variation is a pitchfork bifurcation. In particular, taking the pitchfork bifurcation normal form for reaction-diffusion PDEs is yet another variant within the family of PDEs mentioned above. More generally, the study of this PDE class considering steady states and stability, related to bifurcations due to a parameter is well-understood for the deterministic case. For the stochastic PDE (SPDE) case, the situation is less well-understood and has been studied recently. In this paper we generalize and unify several recent results for SPDE bifurcations. Our generalisation is motivated directly by applications as we introduce in the equation a spatially heterogeneous term and relax the assumptions on the covariance operator that defines the noise. For this spatially heterogeneous SPDE, we prove a finite-time Lyapunov exponent bifurcation result. Furthermore, we extend the theory of early warning signs in our context and we explain, the role of universal exponents between covariance operator warning signs and the lack of finite-time Lyapunov uniformity. Our results are accompanied and cross-validated by numerical simulations.

Identification of the flux function of nonlinear conservation laws with variable parameters

Machine learning methods have in various ways emerged as a useful tool for modeling the dynamics of physical systems in the context of partial differential equations (PDEs). Nonlinear conservation laws (NCLs) of the form ut+f(u)x=0 play a vital role within the family of PDEs. A main challenge with NCLs is that solutions contain discontinuities. That is, one or several jumps of the form (uL(t),uR(t)) with uL≠uR may move in space and time such that information about f(u) in the interval associated with this jump is not present in the observation data. Moreover, the lack of regularity in the solution u(x,t) prevents use of physics informed neural network (PINN) and similar methods. The purpose of this work is to propose a method to identify the nonlinear flux function f(u,β) with variable parameters of an unknown scalar conservation law (∗)ut+f(u,β)x=0with u as the dependent variable and β as the parameter. In a recent work we introduced a framework coined ConsLaw-Net that combines a symbolic multi-layer neural network and an entropy-satisfying discrete scheme to learn the nonlinear, unknown flux function f(u;β) for various fixed β. Learning the flux function with variable parameters, marked as f(u,β), is more challenging since it requires an understanding of the relationship between the variable u and parameter β as well. In this work we demonstrate how to couple ConsLaw-Net to the Linear Regression Neural Network (LRNN) to learn the functional form of the two variable function f(u,β). In addition, ConsLaw-Net is here further developed and made more generic by using a refined discrete scheme combined with a more general symbolic neural network (S-Net). We experimentally demonstrate that the upgraded ConsLaw-Net integrated with LRNN is well-suited for learning tasks, achieving more accurate identification than existing learning approaches when applied to problems that involve general classes of flux functions f(u,β). The investigations of this work are restricted to the class of polynomial, rational functions.

Existence and regularity for global weak solutions to the 𝜆-family water wave equations

In this paper, we prove the global existence of Hölder continuous solutions for the Cauchy problem of a family of partial differential equations, named as λ \lambda -family equations, where λ \lambda is the power of nonlinear wave speed. The λ \lambda -family equations include Camassa-Holm equation ( λ = 1 \lambda =1 ) and Novikov equation ( λ = 2 \lambda =2 ) modelling water waves, where solutions generically form finite time cusp singularities, or, in other words, show wave breaking phenomenon. The global energy conservative solution we construct is Hölder continuous with exponent 1 − 1 2 λ 1- \frac {1}{2\lambda } . The existence result also paves the way for the future study on uniqueness and Lipschitz continuous dependence.

Constructing and Predicting Solutions for Different Families of Partial Differential Equations: A Reliable Algorithm

The construction of mathematical models for different phenomena, and developing their solutions, are critical issues in science and engineering. Among many, the Buckmaster and Korteweg-de Vries (KdV) models are very important due to their ability of capturing different physical situations such as thin film flows and waves on shallow water surfaces. In this manuscript, a new approach based on the generalized Taylor series and residual function is proposed to predict and analyze Buckmaster and KdV type models. This algorithm estimates convergent series with an easy-to-use way of finding solution components through symbolic computation. The proposed algorithm is tested against the Buckmaster and KdV equations, and the results are compared with available solutions in the literature. At first, proposed algorithm is applied to Buckmaster-type linear and nonlinear equations, and attained the closed-form solutions. In the next phase, the proposed algorithm is applied to highly nonlinear KdV equations (namely, classical, modified, and generalized KdV) and approximate solutions are obtained. Simulations of the test problems clearly reassert the dominance and capability of the proposed methodology in terms of accuracy. Analysis reveals that the projected scheme is reliable, and hence, can be utilized for more complex problems in engineering and the sciences.

Probabilistic solutions of fractional differential and partial differential equations and their Monte Carlo simulations

The work in this paper is four-fold. Firstly, we introduce an alternative approach to solve fractional ordinary differential equations as an expected value of a random time process. Using the latter, we present an interesting numerical approach based on Monte Carlo integration to simulate solutions of fractional ordinary and partial differential equations. Thirdly, we show that this approach allows us to find the fundamental solutions for fractional partial differential equations (PDEs), in which the fractional derivative in time is in the Caputo sense and the fractional in space one is in the Riesz–Feller sense. Lastly, using Riccati equation, we study families of fractional PDEs with variable coefficients which allow explicit solutions. Those solutions connect Lie symmetries to fractional PDEs.

Similarity Transformations and Linearization for a Family of Dispersionless Integrable PDEs

We apply the theory of Lie point symmetries for the study of a family of partial differential equations which are integrable by the hyperbolic reductions method and are reduced to members of the Painlevé transcendents. The main results of this study are that from the application of the similarity transformations provided by the Lie point symmetries, all the members of the family of the partial differential equations are reduced to second-order differential equations, which are maximal symmetric and can be linearized.

On the Double ARA-Sumudu Transform and Its Applications

The main purpose of this work is to present a new double transform called the double ARA-Sumudu transform (DARA-ST). The application of the new double transform to some basic functions and the master properties are introduced. The convolution and existence theorems are also presented and proved. These new results are implemented to obtain the solution of partial differential equations (PDEs), integral equations (IEs) and functional equations. We obtain new formulas for solving families of PDEs. The latter ones are used to obtain exact solutions of some familiar PDEs such as the telegraph equation, the advection–diffusion equation, the Klein–Gordon equation and others. Moreover, a simple formula for solving a special kind of integral equations is presented and implemented in some applications. The outcomes show that DARA-ST is useful and efficient in handling such kinds of equations.

A Hamilton–Jacobi PDE Associated with Hydrodynamic Fluctuations from a Nonlinear Diffusion Equation

We study a class of Hamilton–Jacobi partial differential equations in the space of probability measures. In the first part of this paper, we prove comparison principles (implying uniqueness) for this class. In the second part, we establish the existence of a solution and give a representation using a family of partial differential equations with control. A large part of our analysis exploits special structures of the Hamiltonian, which might look mysterious at first sight. However, we show that this Hamiltonian structure arises naturally as limit of Hamiltonians of microscopical models. Indeed, in the third part of this paper, we informally derive the Hamiltonian studied before, in a context of fluctuation theory on the hydrodynamic scale. The analysis is carried out for a specific model of stochastic interacting particles in gas kinetics, namely a version of the Carleman model. We use a two-scale averaging method on Hamiltonians defined in the space of probability measures to derive the limiting Hamiltonian.

On Galilean Invariant and Energy Preserving BBM-Type Equations

We investigate a family of higher-order Benjamin–Bona–Mahony-type equations, which appeared in the course of study towards finding a Galilei-invariant, energy-preserving long wave equation. We perform local symmetry and conservation laws classification for this family of Partial Differential Equations (PDEs). The analysis reveals that this family includes a special equation which admits additional, higher-order local symmetries and conservation laws. We compute its solitary waves and simulate their collisions. The numerical simulations show that their collision is elastic, which is an indication of its S−integrability. This particular PDE turns out to be a rescaled version of the celebrated Camassa–Holm equation, which confirms its integrability.

Simflowny 3: An upgraded platform for scientific modeling and simulation

Simflowny is an open platform which automatically generates efficient parallel code of scientific dynamical models for different simulation frameworks. Here we present major upgrades on this software to support simultaneously a quite generic family of partial differential equations. These equations can be discretized using: (i) standard finite-difference for systems with derivatives up to any order, (ii) High-Resolution-Shock-Capturing methods to deal with shocks and discontinuities of balance law equations, and (iii) particle-based methods. We have improved the adaptive-mesh-refinement algorithms to preserve the convergence order of the numerical methods, which is a requirement for improving scalability. Finally, we have also extended our graphical user interface (GUI) to accommodate these and future families of equations. This paper summarizes the formal representation and implementation of these new families, providing several validation results. Program summaryProgram Title: SimflownyCPC Library link to program files:https://doi.org/10.17632/g9mcw8s64f.2Licensing provisions: Apache License, 2.0Programming language: Java, C++ and JavaScriptJournal Reference of previous version: Comput. Phys. Comm. 184 (2013) 2321–2331, Comput. Phys. Comm. 229 (2018), 170–181Does the new version supersede the previous version?: YesReasons for the new version: Additional featuresSummary of revisions: Expanded support for Partial Differential Equations, meshless particles and advanced Adaptive Mesh Refinement.Nature of problem: Simflowny generates numerical simulation code for a wide range of models.Solution method: Any discretization scheme based on either Finite Volume Methods, Finite Difference Methods, or meshless methods for Partial Differential Equations.Additional comments: Simflowny runs in any computer with Docker [1]. Installation details can be checked in the documentation of Simflowny [2]. It can also be compiled from scratch on any Linux system, provided dependences are properly installed as indicated in the documentation. The generated code runs on any Linux platform ranging from personal workstations to clusters and parallel supercomputers. The software architecture is easily extensible for future additional model families and simulation frameworks. Full documentation is available in the wiki home of the Simflowny project [2].

Nonlocal Effects to Neutron Diffusion Equation in a Nuclear Reactor

In this study, a nonlocal approach to neutron diffusion equation with a memory is constructed in terms of moments of the displacement kernel with a modified geometric buckling. This approach leads to a family of partial differential equations which belong to the class of Fisher-Kolmogorov and Swift-Hohenberg equations. The stability of the problem depends on the signs of the second and fourth moments. The energy is a conserved quantity along orbits and a constant of integration is obtained. It was observed that the buckling is affected by the types of the kernel moment and for an explicit symmetric kernel, the ratio between the maximum and the average flux for a slab reactor is less than the ratio obtained using the conventional local diffusion equation, a result which is motivating technically in nuclear reactor engineering.

PDE Evolutions for M-Smoothers in One, Two, and Three Dimensions

Local M-smoothers are interesting and important signal and image processing techniques with many connections to other methods. In our paper, we derive a family of partial differential equations (PDEs) that result in one, two, and three dimensions as limiting processes from M-smoothers which are based on local order-p means within a ball the radius of which tends to zero. The order p may take any nonzero value $$>-1$$ , allowing also negative values. In contrast to results from the literature, we show in the space-continuous case that mode filtering does not arise for $$p \rightarrow 0$$ , but for $$p \rightarrow -1$$ . Extending our filter class to p-values smaller than $$-1$$ allows to include, e.g. the classical image sharpening flow of Gabor. The PDEs we derive in 1D, 2D, and 3D show large structural similarities. Since our PDE class is highly anisotropic and may contain backward parabolic operators, designing adequate numerical methods is difficult. We present an $$L^\infty $$ -stable explicit finite difference scheme that satisfies a discrete maximum–minimum principle, offers excellent rotation invariance, and employs a splitting into four fractional steps to allow larger time step sizes. Although it approximates parabolic PDEs, it consequently benefits from stabilisation concepts from the numerics of hyperbolic PDEs. Our 2D experiments show that the PDEs for $$p<1$$ are of specific interest: Their backward parabolic term creates favourable sharpening properties, while they appear to maintain the strong shape simplification properties of mean curvature motion.

Boundary layer expansions for initial value problems with two complex time variables

We study a family of partial differential equations in the complex domain, under the action of a complex perturbation parameter ϵ. We construct inner and outer solutions of the problem and relate them to asymptotic representations via Gevrey asymptotic expansions with respect to ϵ in adequate domains. The asymptotic representation leans on the cohomological approach determined by the Ramis–Sibuya theorem.

On the existence and uniqueness of exponentially harmonic maps and some related problems

The family of partial differential equations −eΔu − 2Δ∞u = 0 (e > 0) is studied in a bounded domain Ω for given boundary data. In the case where e = 1, which is closely related to the study of exponentially harmonic maps, we establish existence and uniqueness of a classical solution as the unique minimizer in a closed subset of an Orlicz–Sobolev space of the appropriate energy functional associated to this problem—the integral over Ω of the exponential energy density $$u \mapsto {1 \over 2}\exp \left( {{{\left| {\nabla u} \right|}^2}} \right)$$ . We also explore the connections between the classical solutions of these problems and infinity harmonic and harmonic maps by studying the limiting behavior of the solutions as e → 0+ and e → ∞, respectively. In the former case, we recover a result of Evans and Yu [6].

An extension of Borel–Laplace methods and monomial summability

In this paper we will show that monomial summability can be characterized using Borel–Laplace like integral transformations depending of a parameter 0<s<1. We will apply this result to prove 1-summability in a monomial of formal solutions of a family of partial differential equations.

Corporate dynamics of systems of logistic delay equations with large delay control

A system of two logistic equations with delay coupled by delayed control has been considered. It has been shown that, in the case of a fairly large delay control coefficient, the problem of the dynamics of the initial systems has been reduced to investigating the nonlocal dynamics of special families of partial differential equations that do not contain small and large parameters. New interesting dynamic phenomena are discovered based on the results of numerical analysis. Systems of three logistic delay equations with two types of diffusion relations have been considered. Special families of partial differential equations that do not contain small and large parameters have also been constructed for each of these systems. The research results for the dynamic properties of the original equations have been presented. It has been shown that the difference in the dynamics of the considered systems of three equations may be of a fundamental nature.

An ADER-type scheme for a class of equations arising from the water-wave theory

In this work we propose a numerical strategy to solve a family of partial differential equations arising from the water-wave theory. These problems may contain four terms; a source which is an algebraic function of the solution, a convective part involving first order spatial derivatives of the solution, a diffusive part involving second order spatial derivatives and the transient part. Unlike partial differential equations of hyperbolic or parabolic type, where the transient part is the time derivative of the solution, here the transient part can contain mixed time and space derivatives.In [Zambra et al. International Journal for Numerical Methods in Engineering 89(2):227-240, 2012], the authors proposed a globally implicit strategy to solve the Richards equation. In that case, transient terms consisted of algebraic expressions of the solution. Motivated by this work, we propose a one-step finite volume method to deal with problems in which transient terms are differential operators. Here, a locally implicit formulation is investigated, which is based on the ADER philosophy. The scheme is divided in three steps: i) a polynomial reconstruction of the data; ii) solutions to Generalized Riemann Problems (GRP); iii) the solution of differential problems. Note that steps i) and ii), are those of conventional ADER schemes for conservation laws. Advantages of the present approach include the possibility to construct high-order approximations in both space and time, for which existing methodologies for hyperbolic problems can be applied. The differential problems associated to the transient term can be non-linear and numerical strategies can be adopted to deal wit it. Convergence of the scheme is proved rigorously and an empirical convergence rates assessment is carried out in order to illustrate the high space and time accuracy of the present scheme.