Kuramoto-Sivashinsky Equation Research Articles

Scalable computation of energy functions for nonlinear balanced truncation

Nonlinear balanced truncation is a model order reduction technique that reduces the dimension of nonlinear systems in a manner that accounts for either open- or closed-loop observability and controllability aspects of the system. A computational challenges that has so far prevented its deployment on large-scale systems is that the energy functions required for characterization of controllability and observability are solutions of various high-dimensional Hamilton–Jacobi–(Bellman) equations, which are computationally intractable in high dimensions. This work proposes a unifying and scalable approach to this challenge by considering a Taylor-series-based approximation to solve a class of parametrized Hamilton–Jacobi–Bellman equations that are at the core of nonlinear balancing. The value of a formulation parameter provides either open-loop balancing or a variety of closed-loop balancing options. To solve for the coefficients of Taylor-series approximations to the energy functions, the presented method derives a linear tensor system and heavily utilizes it to numerically solve structured linear systems with billions of unknowns. The strength and scalability of the algorithm is demonstrated on two semi-discretized partial differential equations, namely the Burgers and the Kuramoto–Sivashinsky equations.

On the Nature of Local Bifurcations of the Kuramoto–Sivashinsky Equation in Various Domains

On the Nature of Local Bifurcations of the Kuramoto–Sivashinsky Equation in Various Domains

Bifurcations of Spatially Inhomogeneous Solutions in a Modified Version of the Kuramoto–Sivashinsky Equation

Bifurcations of Spatially Inhomogeneous Solutions in a Modified Version of the Kuramoto–Sivashinsky Equation

An invariance constrained deep learning network for partial differential equation discovery

The discovery of partial differential equations (PDEs) from datasets has attracted increased attention. However, the discovery of governing equations from sparse data with high noise is still very challenging due to the difficulty of derivatives computation and the disturbance of noise. Moreover, the selection principles for the candidate library to meet physical laws need to be further studied. The invariance is one of the fundamental laws for governing equations. In this study, we propose an invariance constrained deep learning network (ICNet) for the discovery of PDEs. Considering that temporal and spatial translation invariance (Galilean invariance) is a fundamental property of physical laws, we filter the candidates that cannot meet the requirement of the Galilean transformations. Subsequently, we embedded the fixed and possible terms into the loss function of neural network, significantly countering the effect of sparse data with high noise. Then, by filtering out redundant terms without fixing learnable parameters during the training process, the governing equations discovered by the ICNet method can effectively approximate the real governing equations. We select the one-dimensional (1D) Kuramoto–Sivashinsky equation, the two-dimensional (2D) Burgers equation, the equation of two-dimensional channel flow over an obstacle, and the equation of three-dimensional (3D) intracranial aneurysm as examples to verify the superiority of the ICNet for fluid mechanics. Furthermore, we extend similar invariance methods to the discovery of wave equation (Lorentz Invariance) and verify it through single and coupled Klein–Gordon equation. The results show that the ICNet method with physical constraints exhibits excellent performance in governing equations discovery from sparse and noisy data.

DEFM: Delay-embedding-based forecast machine for time series forecasting by spatiotemporal information transformation.

Making accurate forecasts for a complex system is a challenge in various practical applications. The major difficulty in solving such a problem concerns nonlinear spatiotemporal dynamics with time-varying characteristics. Takens' delay embedding theory provides a way to transform high-dimensional spatial information into temporal information. In this work, by combining delay embedding theory and deep learning techniques, we propose a novel framework, delay-embedding-based forecast Machine (DEFM), to predict the future values of a target variable in a self-supervised and multistep-ahead manner based on high-dimensional observations. With a three-module spatiotemporal architecture, the DEFM leverages deep neural networks to effectively extract both the spatially and temporally associated information from the observed time series even with time-varying parameters or additive noise. The DEFM can accurately predict future information by transforming spatiotemporal information to the delay embeddings of a target variable. The efficacy and precision of the DEFM are substantiated through applications in three spatiotemporally chaotic systems: a 90-dimensional (90D) coupled Lorenz system, the Lorenz 96 system, and the Kuramoto-Sivashinsky equation with inhomogeneity. Additionally, the performance of the DEFM is evaluated on six real-world datasets spanning various fields. Comparative experiments with five prediction methods illustrate the superiority and robustness of the DEFM and show the great potential of the DEFM in temporal information mining and forecasting.

Dynamic behaviour and semi-analytical solution of nonlinear fractional-order Kuramoto–Sivashinsky equation

Dynamic behaviour and semi-analytical solution of nonlinear fractional-order Kuramoto–Sivashinsky equation

Investigation of facet evolution on Si surfaces bombarded with Xe ions

This study investigates the formation of facets on Si surface under Xe ion irradiation using an ion energy of 0.5 keV. By examining the effects of ion incidence angle (60° –85°), fluence (4.5 × 1018 to 1.35 × 1019 ions/cm2), and temperature (RT to 200 ◦C), we explore the evolution of facets. The surface roughness displays a distinct trend, reaching its peak when the ion incidence angle is 80°, which indicates the formation of faceted structures due to a sudden change in roughness. Additionally, temperature studies highlight the important role of temperature in enhancing facet arrangement. To support experimental findings, numerical simulation using Anisotropic Kuramoto-Sivashinsky (AKS) equation is employed. These simulations provide valuable insights into the dynamics of facet evolution, allowing us to better understand how curvature-dependent sputtering yield, dispersion, and diffusion collectively influence the formation and morphology of facets on the Si surface under Xe ion irradiation.

Thenovel numerical solutions for conformable fractional Kuramoto-Sivashinsky equations by using Cq-HATM and CHPETM

The present study used the novel conformable fractional methods to examine the nonlinear Kuramoto-Sivashinsky equations of the conformable fractional derivative. The Kuramoto-Sivashinsky equation is a mathematical model employed to elucidate a diverse range of chemical and physical phenomena, encompassing chemical reaction-diffusion processes, plasma instabilities, viscous flow issues, flame front propagation, and magnetized plasmas. The present inquiry provides an examination of convergence and error for the future scheme. The conformable q-homotopy analysis transform method (Cq-HATM) provides h-curves that illustrate the convergence range of the series solution achieved. In order to validate the effectiveness and suitability of the Cq-HATM, we examine separate instances. In this study, we introduce an application that demonstrates the potential benefits and effectiveness of the proposed approach. Furthermore, an error analysis is performed in order to verify the accuracy of the scheme. Numerical simulations are conducted to validate the precision of the forthcoming approach. The findings obtained from the numerical and graphical analyses are presented in this study. The method provided in this study showcases a notable degree of computational precision and ease in examining and resolving intricate phenomena related to conformable fractional nonlinear partial differential equations within the domains of science and technology.

Iterative ensemble smoothers for data assimilation in coupled nonlinear multiscale models

Abstract This paper identifies and explains particular differences and properties of adjoint-free iterative ensemble methods initially developed for parameter estimation in petroleum models. The aim is to demonstrate the methods’ potential for sequential data assimilation in coupled and multiscale unstable dynamical systems. For this study, we have introduced a new nonlinear and coupled multiscale model based on two Kuramoto-Sivashinsky equations operating on different scales where a coupling term relaxes the two model variables towards each other. This model provides a convenient testbed for studying data assimilation in highly nonlinear and coupled multiscale systems. We show that the model coupling leads to cross-covariance between the two models’ variables, allowing for a combined update of both models. The measurements of one model’s variable will also influence the other and contribute to a more consistent estimate. Secondly, the new model allows us to examine the properties of iterative ensemble smoothers and assimilation updates over finite-length assimilation windows. We discuss the impact of varying the assimilation windows’ length relative to the model’s predictability time scale. Furthermore, we show that iterative ensemble smoothers significantly improve the solution’s accuracy compared to the standard ensemble-Kalman-filter update. Results and discussions provide an enhanced understanding of the ensemble methods’ potential implementation and use in operational weather and climate-prediction systems.

Data-driven modeling and forecasting of chaotic dynamics on inertial manifolds constructed as spectral submanifolds.

We present a data-driven and interpretable approach for reducing the dimensionality of chaotic systems using spectral submanifolds (SSMs). Emanating from fixed points or periodic orbits, these SSMs are low-dimensional inertial manifolds containing the chaotic attractor of the underlying high-dimensional system. The reduced dynamics on the SSMs turn out to predict chaotic dynamics accurately over a few Lyapunov times and also reproduce long-term statistical features, such as the largest Lyapunov exponents and probability distributions, of the chaotic attractor. We illustrate this methodology on numerical data sets including delay-embedded Lorenz and Rössler attractors, a nine-dimensional Lorenz model, a periodically forced Duffing oscillator chain, and the Kuramoto-Sivashinsky equation. We also demonstrate the predictive power of our approach by constructing an SSM-reduced model from unforced trajectories of a buckling beam and then predicting its periodically forced chaotic response without using data from the forced beam.

Nonintrusive Operator Inference for Chaotic Systems

This work explores the physics-driven machine learning technique Operator Inference (OpInf) for predicting the state of chaotic dynamical systems. OpInf provides a non-intrusive approach to infer approximations of polynomial operators in reduced space without having access to the full order operators appearing in discretized models. Datasets for the physics systems are generated using conventional numerical solvers and then projected to a low-dimensional space via Principal Component Analysis (PCA). In latent space, a least-squares problem is set to fit a quadratic polynomial operator, which is subsequently employed in a time-integration scheme in order to produce extrapolations in the same space. Once solved, the inverse PCA operation is applied to reconstruct the extrapolations in the original space. The quality of the OpInf predictions is assessed via the Normalized Root Mean Squared Error (NRMSE) metric from which the Valid Prediction Time (VPT) is computed. Numerical experiments considering the chaotic systems Lorenz 96 and the Kuramoto-Sivashinsky equation show promising forecasting capabilities of the OpInf reduced order models with VPT ranges that outperform state-of-the-art machine learning (ML) methods such as backpropagation and reservoir computing recurrent neural networks [1], as well as Markov neural operators [2].

Constructing polynomial libraries for reservoir computing in nonlinear dynamical system forecasting.

Reservoir computing is an effective model for learning and predicting nonlinear and chaotic dynamical systems; however, there remains a challenge in achieving a more dependable evolution for such systems. Based on the foundation of Koopman operator theory, considering the effectiveness of the sparse identification of nonlinear dynamics algorithm to construct candidate nonlinear libraries in the application of nonlinear data, an alternative reservoir computing method is proposed, which creates the linear Hilbert space of the nonlinear system by including nonlinear terms in the optimization process of reservoir computing, allowing for the application of linear optimization. We introduce an implementation that incorporates a polynomial transformation of arbitrary order when fitting the readout matrix. Constructing polynomial libraries with reservoir-state vectors as elements enhances the nonlinear representation of reservoir states and more easily captures the complexity of nonlinear systems. The Lorenz-63 system, the Lorenz-96 system, and the Kuramoto-Sivashinsky equation are used to validate the effectiveness of constructing polynomial libraries for reservoir states in the field of state-evolution prediction of nonlinear and chaotic dynamical systems. This study not only promotes the theoretical study of reservoir computing, but also provides a theoretical and practical method for the prediction of nonlinear and chaotic dynamical system evolution.

Retracted: Analysis of Fuzzy Kuramoto-Sivashinsky Equations under a Generalized Fuzzy Fractional Derivative Operator

Retracted: Analysis of Fuzzy Kuramoto-Sivashinsky Equations under a Generalized Fuzzy Fractional Derivative Operator

A higher order stable numerical approximation for time‐fractional non‐linear Kuramoto–Sivashinsky equation based on quintic B$$ \mathfrak{B} $$‐spline

This article deals with designing and analyzing a higher order stable numerical analysis for the time‐fractional Kuramoto–Sivashinsky (K‐S) equation, which is a fourth‐order non‐linear equation. The fractional derivative of order present in the considered problem is taken into Caputo sense and approximated using the scheme. In space direction, the discretization process uses quintic ‐spline functions to approximate the derivatives and the solution of the problem. The present approach is unconditionally stable and is convergent with rate of accuracy , where and denote the space and time step sizes, respectively. We have also noted that the linearized version of the K‐S equation leads the rate of accuracy to . The present approach is also highly effective for the time‐fractional Burgers' equation. We have shown that the present approach provides better accuracy than the scheme with the same computational cost for several linear/non‐linear problems, with classical as well as fractional time derivatives.

Kuramoto-Sivashinsky equation: Numerical solution using two quintic B-splines and differential quadrature method

This paper aims to obtain the numerical solution of a one-dimension fourth-order Kuramoto-Sivashinsky (KS) equation, which has application in the context of flame propagation. This study combines the implementation of two new mechanisms quintic Uniform Algebraic Hyperbolic (QUAH) tension B-spline and quintic Uniform Algebraic Trigonometric (QUAT) tension B-spline with a renowned method known as Differential Quadrature Method (DQM). DQM is used to approximate the derivatives. Then, to obtain the weighting coefficients both quintic splines QUAH and QUAT are implemented with DQM and the original boundary value problem was then transformed into an ordinary differential equation. The Runge-Kutta 43 technique was used to solve the resultant ODE. With this approach, the outcomes are precise and close to the exact solution. The differential quadrature technique has a significant advantage over the previous approaches because it prevents the perturbation in order to find the better results for the given nonlinear equations. The accuracy of the suggested method is then illustrated numerically resolving four test problems and calculating the error norms L2 and L∞. Calculated results are displayed graphically and in tabular format for quick and simple access. This paper introduces an original work, to the best of the author’s knowledge. The obtained solution and graphical observations show that QUAH and QUAT tension B-splines with DQM is a strong and dependable methods for estimating nonlinear problem solutions. Quintic B-splines are higher-order interpolation method compared to lower-degree B-splines, which results in more accurate approximations, and provides smooth and continuous representations of functions. Moreover, DQM when combined with quintic B-splines, can help reduce numerical dispersion, and can be applied to a wide range of problems.

Front instabilities in the presence of convection due to thermal and compositional gradients.

Reaction fronts separate fluids of different densities due to thermal and compositional gradients that may lead to convection. The stability of convectionless flat fronts propagating in the vertical direction depends not only on fluid properties but also in the dynamics of a front evolution equation. In this work, we analyze fronts described by the Kuramoto-Sivashinsky (KS) equation coupled to hydrodynamics. Without density gradients, the KS equation has a flat front solution that is unstable to perturbations of long wavelengths. Buoyancy enhances this instability if a fluid of lower density is underneath a denser fluid. In the reverse situation, with the denser fluid underneath, the front can be stabilized with appropriate thermal and compositional gradients. However, in this situation, a different instability develops for large enough thermal gradients. We also solve numerically the nonlinear KS equation coupled to the Navier-Stokes equations to analyze the front propagation in two-dimensional rectangular domains. As convection takes place, the reaction front curves, increasing its velocity.

Reliable iterative techniques for solving the KS equation arising in fluid flow

In this study, we examine the Kuramoto-Sivashinsky equation which is a nonlinear model that describes several physical and chemical events arising in fluid flow. The approximate analytical solution for the fractional KS (FKS) problem is calculated using the Temimi-Ansari method (TAM) and the natural decomposition method (NDM). The projected procedure (NDM) combines the adomian decomposition method with the natural transform. Each technique can deal with nonlinear terms without making any assumptions. The methodologies under consideration provide � � -curves that display the convergence window of the power series solution that approaches the exact solution. We explore two distinct examples to confirm the efficiency and applicability of the proposed strategies. The acquired outcomes are compared numerically with the q-homotopy analysis transform method (q-HATM). The numerical investigation is carried out to validate the precision and dependability of the approaches under consideration. Additionally, the nature of the outcomes gained has been displayed in a different order. The obtained results show that the proposed techniques are highly efficient and simple to use to analyze the behavior of other nonlinear models. Keywords: Kuramoto-Sivashinsky equation, Laplace transform, natural decomposition method (NDM), Temimi-Ansari method (TAM)

Analytical solutions to time-space fractional Kuramoto-Sivashinsky Model using the integrated Bäcklund transformation and Riccati-Bernoulli sub-ODE method

<abstract><p>This paper solves an example of a time-space fractional Kuramoto-Sivashinsky (KS) equation using the integrated Bäcklund transformation and the Riccati-Bernoulli sub-ODE method. A specific version of the KS equation with power nonlinearity of a given degree is examined. Using symbolic computation, we find new analytical solutions to the current problem for modeling many nonlinear phenomena that are described by this equation, like how the flame front moves back and forth, how fluids move down a vertical wall, or how chemical reactions happen in a uniform medium while they oscillate uniformly across space. In the field of mathematical physics, the Riccati-Bernoulli sub-ODE approach is shown to be a valuable tool for producing a variety of single solutions.</p></abstract>

About classical solutions for high order conserved Kuramoto-Sivashinsky type equation

About classical solutions for high order conserved Kuramoto-Sivashinsky type equation

Continuity in expectation of odd random attractors for stochastic Kuramoto-Sivashinsky equations

Continuity in expectation of odd random attractors for stochastic Kuramoto-Sivashinsky equations