Kuramoto-Sivashinsky Equation Research Articles

Data-driven control of spatiotemporal chaos with reduced-order neural ODE-based models and reinforcement learning

Deep reinforcement learning (RL) is a data-driven method capable of discovering complex control strategies for high-dimensional systems, making it promising for flow control applications. In particular, the present work is motivated by the goal of reducing energy dissipation in turbulent flows, and the example considered is the spatiotemporally chaotic dynamics of the Kuramoto–Sivashinsky equation (KSE). A major challenge associated with RL is that substantial training data must be generated by repeatedly interacting with the target system, making it costly when the system is computationally or experimentally expensive. We mitigate this challenge in a data-driven manner by combining dimensionality reduction via an autoencoder with a neural ODE framework to obtain a low-dimensional dynamical model from just a limited data set. We substitute this data-driven reduced-order model (ROM) in place of the true system during RL training to efficiently estimate the optimal policy, which can then be deployed on the true system. For the KSE actuated with localized forcing (‘jets’) at four locations, we demonstrate that we are able to learn a ROM that accurately captures the actuated dynamics as well as the underlying natural dynamics just from snapshots of the KSE experiencing random actuations. Using this ROM and a control objective of minimizing dissipation and power cost, we extract a control policy from it using deep RL. We show that the ROM-based control strategy translates well to the true KSE and highlight that the RL agent discovers and stabilizes an underlying forced equilibrium solution of the KSE system. We show that this forced equilibrium captured in the ROM and discovered through RL is related to an existing known equilibrium solution of the natural KSE.

Stabilization by switching of parabolic PDEs with spatially scheduled actuators and sensors

This paper addresses a switched sampled-data control design for stabilization of Kuramoto-Sivashinsky equation under the Dirichlet/periodic boundary conditions with spatially scheduled actuators. It is supposed that discrete-time point-like or averaged measurements are available. The system is known to be stabilizable by static output-feedback employing several distributed in space actuators and sensors, but is not stabilizable by only one of the actuator–sensor pairs. Does there exist a switching stabilizing static output-feedback such that at all times, only one actuator–sensor pair is active? We give a positive answer and find the appropriate switching sampled-data control law. The proposed switching controller can be implemented either by N actuators and sensors placed in each subdomain (here switching control may reduce the energy that the system spends) or by using one actuator–sensor pair that can move to the active subdomain. For implementation of the control law by moving actuators and sensors, we take into account a moving time by treating it as an additional switching between the open-loop system and the closed-loop switched system. The guidance of active (or mobile) actuators and sensors is provided by using output-dependent switching. Constructive conditions are derived to ensure that the resulting closed-loop system is regionally stable by means of the Lyapunov approach. Numerical example illustrates the efficiency of the method.

The maximum likelihood ensemble smoother for the Kuramoto–Sivashinsky equation

AbstractData assimilation (DA) aims to combine observations/data with a model to maximize the utility of information for obtaining the optimal estimate. The maximum likelihood ensemble filter (MLEF) is a sequential DA method or a filter-type method. Weaknesses of the filter method are assimilating time-integrated observations and estimating empirical parameter estimation. The reason is that the forward model is employed outside of the analysis procedure in this type of DA method. To overcome these weaknesses, the MLEF is now extended as a smoother and the novel maximum likelihood ensemble smoother (MLES) is proposed. The MLES is a smoothing method with variational-like qualities, specifically in the cost function. Rather than using the error information from a single temporal location to solve for the optimal analysis update as done by the MLEF, the MLES can include observations and the forward model within a chosen time window. The newly proposed DA method is first validated by a series of rigorous and thorough performance tests using the Lorenz 96 model. Then, as DA is known to be used extensively to increase the predictability of the commonly chaotic dynamical systems seen in meteorological applications, this study demonstrates the MLES with a model chaotic problem governed by the 1D Kuramoto–Sivashinky (KS) equation. Additionally, the MLES is shown to be an effective method in improving the estimate of uncertain empirical model parameters. The MLES and MLEF are then directly compared and it is shown that the performance of the MLES is adequate and that it is a good candidate for increasing the predictability of a chaotic dynamical system. Future work will focus on an extensive application of the MLES to highly turbulent flows.

Stackelberg-Nash exact controllability for the Kuramoto-Sivashinsky equation with boundary and distributed controls

This paper deals with a multi-objective control problem for the Kuramoto-Sivashinsky equation by following a Stackelberg-Nash strategy. We have a distributed control called Leader, and two boundary controls called Followers, each of them has to act over the equation to influence the behavior of the state in a particular way, by reaching or approaching to many targets at once. To be more precise, the Leader wants to drive the solution to a prescribed target at a final time, and the followers have to minimize some given cost functionals, adapting themselves to what the Leader wants. The main difficulty here is that, since the Followers are in the boundary, the problem turns to be equivalent to prove a partial null controllability result for a system of nonlinear fourth-order equations with boundary coupling terms.

An improved sparse identification of nonlinear dynamics with Akaike information criterion and group sparsity.

A crucial challenge encountered in diverse areas of engineering applications involves speculating the governing equations based upon partial observations. On this basis, a variant of the sparse identification of nonlinear dynamics (SINDy) algorithm is developed. First, the Akaike information criterion (AIC) is integrated to enforce model selection by hierarchically ranking the most informative model from several manageable candidate models. This integration avoids restricting the number of candidate models, which is a disadvantage of the traditional methods for model selection. The subsequent procedure expands the structure of dynamics from ordinary differential equations (ODEs) to partial differential equations (PDEs), while group sparsity is employed to identify the nonconstant coefficients of partial differential equations. Of practical consideration within an integrated frame is data processing, which tends to treat noise separate from signals and tends to parametrize the noise probability distribution. In particular, the coefficients of a species of canonical ODEs and PDEs, such as the Van der Pol, Rössler, Burgers’ and Kuramoto–Sivashinsky equations, can be identified correctly with the introduction of noise. Furthermore, except for normal noise, the proposed approach is able to capture the distribution of uniform noise. In accordance with the results of the experiments, the computational speed is markedly advanced and possesses robustness.

Blowing-up solutions for time-fractional equations on a bounded domain

This paper proposes initial-boundary value problems for time-fractional analogs of Kuramoto-Sivashinsky, Korpusov-Pletner-Sveshnikov, Cahn-Allen, and Hoff equations due to a bounded domain. Adequate conditions for the blowing-up of solutions in limited time of previously mentioned conditions are displayed. The Pohozhaev nonlinear capacity strategy is considered. Illustrative examples are given for each of the investigated equations. (2010) Mathematics Subject Classifications: 34A08; 34B25; 34B15.

Finite-Dimensional Boundary Control of the Linear Kuramoto-Sivashinsky Equation Under Point Measurement With Guaranteed $L^2$-Gain

Finite-dimensional observer-based controller design for PDEs is a challenging problem. Recently, such controllers were introduced for the one dimensional (1D) heat equation, under the assumption that one of the observation or control operators is bounded. This article suggests a constructive method for such controllers for 1D parabolic partial differential equations (PDEs) with both observation and control operators being unbounded. We consider the Kuramoto–Sivashinsky equation under either boundary or in-domain point measurement and boundary actuation in the presence of disturbances in the PDE and measurement. We employ a modal decomposition approach via dynamic extension, using eigenfunctions of a Sturm–Liouville operator. The controller dimension is defined by the number of unstable modes, whereas the observer dimension <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> may be larger. We suggest a direct Lyapunov approach to the full-order closed-loop system, which results in a linear matrix inequality (LMI), for input-to-state stabilization (ISS) and guaranteed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^2$</tex-math></inline-formula> -gain, whose elements and dimension depend on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> . The value of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> and the decay rate are obtained from the LMI. We prove that the LMI is always feasible provided <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> and the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^2$</tex-math></inline-formula> or ISS gains are large enough, thereby obtaining guarantees for our approach. Moreover, for the case of stabilization, we show that feasibility of the LMI for some <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> implies its feasibility for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N+1$</tex-math></inline-formula> . Numerical examples demonstrate the efficiency of the method.

On the Classical Solutions for the Kuramoto-Sivashinsky Equation with Ehrilch-Schwoebel Effects

The Kuramoto-Sivashinsky equation with Ehrilch-Schwoebel effects models the evolution of surface morphology during Molecular Beam Epitaxy growth, provoked by an interplay between deposition of atoms onto the surface and the relaxation of the surface profile through surface diffusion. It consists of a nonlinear fourth order partial differential equation. Using energy methods we prove the well-posedness (i.e., existence, uniqueness and stability with respect to the initial data) of the classical solutions for the Cauchy problem, associated with this equation.

Surface similarity parameter: A new machine learning loss metric for oscillatory spatio-temporal data

Supervised machine learning approaches require the formulation of a loss functional to be minimized in the training phase. Sequential data are ubiquitous across many fields of research, and are often treated with Euclidean distance-based loss functions that were designed for tabular data. For smooth oscillatory data, those conventional approaches lack the ability to penalize amplitude, frequency and phase prediction errors at the same time, and tend to be biased towards amplitude errors. We introduce the surface similarity parameter (SSP) as a novel loss function that is especially useful for training machine learning models on smooth oscillatory sequences. Our extensive experiments on chaotic spatio-temporal dynamical systems indicate that the SSP is beneficial for shaping gradients, thereby accelerating the training process, reducing the final prediction error, increasing weight initialization robustness, and implementing a stronger regularization effect compared to using classical loss functions. The results indicate the potential of the novel loss metric particularly for highly complex and chaotic data, such as data stemming from the nonlinear two-dimensional Kuramoto–Sivashinsky equation and the linear propagation of dispersive surface gravity waves in fluids.

Null Boundary Controllability of the Linear Kuramoto–Sivashinsky Equation with Variable Coefficients

Null Boundary Controllability of the Linear Kuramoto–Sivashinsky Equation with Variable Coefficients

Regularity Criteria for the Kuramoto–Sivashinsky Equation in Dimensions Two and Three

We propose and prove several regularity criteria for the 2D and 3D Kuramoto–Sivashinsky equation, in both its scalar and vector forms. In particular, we examine integrability criteria for the regularity of solutions in terms of the scalar solution phi , the vector solution utriangleq nabla phi , as well as the divergence text {div}(u)=Delta phi , and each component of u and nabla u. We also investigate these criteria computationally in the 2D case, and we include snapshots of solutions for several quantities of interest that arise in energy estimates.

Solitary Wave Propagation of the Generalized Kuramoto-Sivashinsky Equation in Fragmented Porous Media

Solitary Wave Propagation of the Generalized Kuramoto-Sivashinsky Equation in Fragmented Porous Media

A single bounded input‐feedback control to the generalized Korteweg–de Vries–Burgers–Kuramoto–Sivashinsky equation

The generalized Korteweg–de Vries–Burgers–Kuramoto–Sivashinsky equation (GKdVBKS) is a nonlinear partial differential equation that models propagation of waves in a thick elastic tube filled with a viscous fluid. First, the dynamics of the GKdVBKS equation that depend on the physical parameters and , where is a positive parameter and is a real number, are presented. We show that the set of constant equilibrium solutions is unstable when , Lyapunov stable when , and globally exponentially stable when . Second, we show that the controlled GKdVBKS equation has a unique strong solution. Then, we synthesize a single bounded input‐feedback controller for the GKdVBKS equation through its linearized system avoiding any spillover. The approach used is valid due to the Fréchet differentiability of the nonlinear ‐semigroup generated by the nonlinear GKdVBKS equation with its derivative being the linear ‐semigroup generated by the linearized system. Finally, numerical results for different values of the parameters of the equation are presented to illustrate the developed theory.

Ensemble Kalman inversion for sparse learning of dynamical systems from time-averaged data

Enforcing sparse structure within learning has led to significant advances in the field of data-driven discovery of dynamical systems. However, such methods require access not only to timeseries of the state of the dynamical system, but also to the time derivative. In many applications, the data are available only in the form of time-averages such as moments and autocorrelation functions. We propose a sparse learning methodology to discover the vector fields defining a (possibly stochastic or partial) differential equation, using only time-averaged statistics. Such a formulation of sparse learning naturally leads to a nonlinear inverse problem to which we apply the methodology of ensemble Kalman inversion (EKI). EKI is chosen because it may be formulated in terms of the iterative solution of quadratic optimization problems; sparsity is then easily imposed. We then apply the EKI-based sparse learning methodology to various examples governed by stochastic differential equations (a noisy Lorenz 63 system), ordinary differential equations (Lorenz 96 system and coalescence equations), and a partial differential equation (the Kuramoto-Sivashinsky equation). The results demonstrate that time-averaged statistics can be used for data-driven discovery of differential equations using sparse EKI. The proposed sparse learning methodology extends the scope of data-driven discovery of differential equations to previously challenging applications and data-acquisition scenarios.

Fractional View Analysis of Kuramoto–Sivashinsky Equations with Non-Singular Kernel Operators

In this article, we investigate the nonlinear model describing the various physical and chemical phenomena named the Kuramoto–Sivashinsky equation. We implemented the natural decomposition method, a novel technique, mixed with the Caputo–Fabrizio (CF) and Atangana–Baleanu deriavatives in Caputo manner (ABC) fractional derivatives for obtaining the approximate analytical solution of the fractional Kuramoto–Sivashinsky equation (FKS). The proposed method gives a series form solution which converges quickly towards the exact solution. To show the accuracy of the proposed method, we examine three different cases. We presented proposed method results by means of graphs and tables to ensure proposed method validity. Further, the behavior of the achieved results for the fractional order is also presented. The results we obtain by implementing the proposed method shows that our technique is extremely efficient and simple to investigate the behaviour of nonlinear models found in science and technology.

Global regular solutions for the multi-dimensional Kuramoto-Sivashinsky equation posed on smooth domains

Initial-boundary value problems for the $n$-dimensional ($n$ is a natural number from the interval [2,7]) Kuramoto-Sivashinsky equation posed on smooth bounded domains in $\mathbb{R}^n$ were considered. The existence and uniqueness of global regular solutions as well as their exponential decay have been considered.

An approximate inertial manifold (AIM) based closure for turbulent flows

A closure model for turbulent flows is developed based on a dynamical system theory. An appropriately discretized formulation of the governing equations is considered for this process. The key ingredient is an approximation of the system’s attractor, where all the trajectories in phase space are confined. This approximate inertial manifold based approach provides a path to track trajectories of the system in a lower-dimensional subspace. Unlike conventional coarse-graining approaches, the turbulent field is decomposed into resolved and unresolved dynamics using the properties of the governing equations. The novelty of the approach relies on the reconstruction of the unresolved field constrained by the governing equations. A posteriori tests for homogeneous isotropic turbulence and the Kuramoto–Sivashinsky equation show promising results for considerable dimension reduction with strong convergence properties. The proposed model outperforms the dynamic Smagorinsky model, and the computational overhead is competitive with similar approaches.

Data-driven reduced-order modeling of spatiotemporal chaos with neural ordinary differential equations.

Dissipative partial differential equations that exhibit chaotic dynamics tend to evolve to attractors that exist on finite-dimensional manifolds. We present a data-driven reduced-order modeling method that capitalizes on this fact by finding a coordinate representation for this manifold and then a system of ordinary differential equations (ODEs) describing the dynamics in this coordinate system. The manifold coordinates are discovered using an undercomplete autoencoder-a neural network (NN) that reduces and then expands dimension. Then, the ODE, in these coordinates, is determined by a NN using the neural ODE framework. Both of these steps only require snapshots of data to learn a model, and the data can be widely and/or unevenly spaced. Time-derivative information is not needed. We apply this framework to the Kuramoto-Sivashinsky equation for domain sizes that exhibit chaotic dynamics with again estimated manifold dimensions ranging from 8 to 28. With this system, we find that dimension reduction improves performance relative to predictions in the ambient space, where artifacts arise. Then, with the low-dimensional model, we vary the training data spacing and find excellent short- and long-time statistical recreation of the true dynamics for widely spaced data (spacing of ∼ 0.7 Lyapunov times). We end by comparing performance with various degrees of dimension reduction and find a "sweet spot" in terms of performance vs dimension.

Hierarchical deep learning of multiscale differential equation time-steppers.

Nonlinear differential equations rarely admit closed-form solutions, thus requiring numerical time-stepping algorithms to approximate solutions. Further, many systems characterized by multiscale physics exhibit dynamics over a vast range of timescales, making numerical integration expensive. In this work, we develop a hierarchy of deep neural network time-steppers to approximate the dynamical system flow map over a range of time-scales. The model is purely data-driven, enabling accurate and efficient numerical integration and forecasting. Similar ideas can be used to couple neural network-based models with classical numerical time-steppers. Our hierarchical time-stepping scheme provides advantages over current time-stepping algorithms, including (i) capturing a range of timescales, (ii) improved accuracy in comparison with leading neural network architectures, (iii) efficiency in long-time forecasting due to explicit training of slow time-scale dynamics, and (iv) a flexible framework that is parallelizable and may be integrated with standard numerical time-stepping algorithms. The method is demonstrated on numerous nonlinear dynamical systems, including the Van der Pol oscillator, the Lorenz system, the Kuramoto–Sivashinsky equation, and fluid flow pass a cylinder; audio and video signals are also explored. On the sequence generation examples, we benchmark our algorithm against state-of-the-art methods, such as LSTM, reservoir computing and clockwork RNN.This article is part of the theme issue ‘Data-driven prediction in dynamical systems’.

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

This paper evaluates a semianalytical strategy combined with a novel fuzzy integral transformation and an iterative method inside the fuzziness concept known as the new iterative transform method. Additionally, we apply the abovementioned technique to the fractional fuzzy Kuramoto-Sivashinsky equations with g H -differentiability by employing various initial conditions. Numerous algebraic properties of the fuzzy fractional derivative Atangana-Baleanu operator are illustrated concerning the Shehu transformation to demonstrate their utility. Additionally, a general technique for Atangana-Baleanu fuzzy fractional derivatives is proposed in the sense of Caputo. It is important to note that the purpose of the suggested fuzziness technique is to establish the efficiency and accuracy of analytical solution to nonlinear fuzzy fractional partial differential equations that emerge in complex and physical structures.