Manifold Of Solutions Research Articles

From chimeras to extensive chaos in networks of heterogeneous Kuramoto oscillator populations.

Populations of coupled oscillators can exhibit a wide range of complex dynamical behavior, from complete synchronization to chimera and chaotic states. We can, thus, expect complex dynamics to arise in networks of such populations. Here, we analyze the dynamics of networks of populations of heterogeneous mean-field coupled Kuramoto-Sakaguchi oscillators and show that the instability that leads to chimera states in a simple two-population model also leads to extensive chaos in large networks of coupled populations. Formally, the system consists of a complex network of oscillator populations whose mesoscopic behavior evolves according to the Ott-Antonsen equations. By considering identical parameters across populations, the system contains a manifold of homogeneous solutions where all populations behave identically. Stability analysis of these homogeneous states provided by the master stability function formalism shows that non-trivial dynamics might emerge on a wide region of the parameter space for arbitrary network topologies. As examples, we first revisit the two-population case and provide a complete bifurcation diagram. Then, we investigate the emergent dynamics in large ring and Erdös-Rényi networks. In both cases, transverse instabilities lead to extensive space-time chaos, i.e., irregular regimes whose complexity scales linearly with the system size. Our work provides a unified analytical framework to understand the emergent dynamics of networks of oscillator populations, from chimera states to robust high-dimensional chaos.

Compositional Sparsity, Approximation Classes, and Parametric Transport Equations

Abstract Approximating functions of a large number of variables poses particular challenges often subsumed under the term “Curse of Dimensionality” (CoD). Unless the approximated function exhibits a very high level of smoothness the CoD can be avoided only by exploiting some typically hidden structural sparsity. In this paper we propose a general framework for new model classes of functions in high dimensions. They are based on suitable notions of compositional dimension-sparsity quantifying, on a continuous level, approximability by compositions with certain structural properties. In particular, this describes scenarios where deep neural networks can avoid the CoD. The relevance of these concepts is demonstrated for solution manifolds of parametric transport equations. For such PDEs parameter-to-solution maps do not enjoy the type of high order regularity that helps to avoid the CoD by more conventional methods in other model scenarios. Compositional sparsity is shown to serve as the key mechanism for proving that sparsity of problem data is inherited in a quantifiable way by the solution manifold. In particular, one obtains convergence rates for deep neural network realizations showing that the CoD is indeed avoided.

Explicable hyper-reduced order models on nonlinearly approximated solution manifolds of compressible and incompressible Navier-Stokes equations

Explicable hyper-reduced order models on nonlinearly approximated solution manifolds of compressible and incompressible Navier-Stokes equations

Registration-based nonlinear model reduction of parametrized aerodynamics problems with applications to transonic Euler and RANS flows

We develop a registration-based nonlinear model-order reduction (MOR) method for partial differential equations (PDEs) with applications to transonic Euler and Reynolds-averaged Navier–Stokes (RANS) equations in aerodynamics. These PDEs exhibit discontinuous features, namely shocks, whose location depends on problem configuration parameters, and the associated parametric solution manifold exhibits a slowly decaying Kolmogorov N-width. As a result, conventional linear MOR methods, which use linear reduced approximation spaces, do not yield accurate low-dimensional approximations. We present a registration-based nonlinear MOR method to overcome this challenge. Our formulation builds on the following key ingredients: (i) a geometrically transformable parametrized PDE discretization; (ii) localized spline-based parametrized transformations which warp the domain to align discontinuities; (iii) an efficient dilation-based shock sensor and metric to compute optimal transformation parameters; (iv) hyperreduction and online-efficient output-based error estimates; and (v) simultaneous transformation and adaptive finite element training. Compared to existing methods in the literature, our formulation is efficiently scalable to larger problems and is equipped with error estimates and hyperreduction. We demonstrate the effectiveness of the method on two-dimensional inviscid and turbulent flows modeled by the Euler and RANS equations, respectively.

Zero‐curvature subconformal structures and dispersionless integrability in dimension five

AbstractWe extend the recent paradigm “Integrability via Geometry” from dimensions 3 and 4 to higher dimensions, relating dispersionless integrability of partial differential equations to curvature constraints of the background geometry. We observe that in higher dimensions on any solution manifold, the symbol defines a vector distribution equipped with a subconformal structure, and the integrability imposes a certain compatibility between them. In dimension 5, we express dispersionless integrability via the vanishing of a certain curvature of this subconformal structure. We also obtain a “master equation” governing all second‐order dispersionless integrable equations in 5D, and count their functional dimension. It turns out that the obtained background geometry is parabolic of the type . We provide its Cartan‐theoretic description and compute the harmonic curvature components via the Kostant theorem. Then, we relate it to 3D projective and 4D conformal geometries via twistor theory, discuss symmetry reductions and nested Lax sequences, as well as give another interpretation of dispersionless integrability in 5D through Levi‐degenerate CR structures in 7D.

Calibration-Based ALE Model Order Reduction for Hyperbolic Problems with Self-Similar Travelling Discontinuities

We propose a novel Model Order Reduction framework that is able to handle solutions of hyperbolic problems characterized by multiple travelling discontinuities. By means of an optimization based approach, we introduce suitable calibration maps that allow us to transform the original solution manifold into a lower dimensional one. The novelty of the methodology is represented by the fact that the optimization process does not require the knowledge of the discontinuities location. The optimization can be carried out simply by choosing some reference control points, thus avoiding the use of some implicit shock tracking techniques, which would translate into an increased computational effort during the offline phase. In the online phase, we rely on a non-intrusive approach, where the coefficients of the projection of the reduced order solution onto the reduced space are recovered by means of an Artificial Neural Network. To validate the methodology, we present numerical results for the 1D Sod shock tube problem, for the 2D double Mach reflection problem, also in the parametric case, and for the triple point problem.

A subspace‐adaptive weights cubature method with application to the local hyperreduction of parameterized finite element models

AbstractThis article is concerned with quadrature/cubature rules able to deal with multiple subspaces of functions, in such a way that the integration points are common for all the subspaces, yet the (nonnegative) weights are tailored to each specific subspace. These subspace‐adaptive weights cubature rules can be used to accelerate computational mechanics applications requiring efficiently evaluating spatial integrals whose integrand function dynamically switches between multiple pre‐computed subspaces. One of such applications is local hyperreduced‐order modeling (HROM), in which the solution manifold is approximately represented as a collection of basis matrices, each basis matrix corresponding to a different region in parameter space. The proposed optimization framework is discrete in terms of the location of the integration points, in the sense that such points are selected among the Gauss points of a given finite element mesh, and the target subspaces of functions are represented by orthogonal basis matrices constructed from the values of the functions at such Gauss points, using the singular value decomposition (SVD). This discrete framework allows us to treat also problems in which the integrals are approximated as a weighted sum of the contribution of each finite element, as in the energy‐conserving sampling and weighting method of C. Farhat and co‐workers. Two distinct solution strategies are examined. The first one is a greedy strategy based on an enhanced version of the empirical cubature method (ECM) developed by the authors elsewhere (we call it the subspace‐adaptive weights ECM, SAW‐ECM for short), while the second method is based on a convexification of the cubature problem so that it can be addressed by linear programming algorithms. We show in a toy problem involving integration of polynomial functions that the SAW‐ECM clearly outperforms the other method both in terms of computational cost and optimality. On the other hand, we illustrate the performance of the SAW‐ECM in the construction of a local HROMs in a highly nonlinear equilibrium problem (large strains regime). We demonstrate that, provided that the subspace‐transition errors are negligible, the error associated to hyperreduction using adaptive weights can be controlled by the truncation tolerances of the SVDs used for determining the basis matrices. We also show that the number of integration points decreases notably as the number of subspaces increases, and that, in the limiting case of using as many subspaces as snapshots, the SAW‐ECM delivers rules with a number of integration points only dependent on the intrinsic dimensionality of the solution manifold and the degree of overlapping required to avoid subspace‐transition errors. The Python source codes of the proposed SAW‐ECM are openly accessible in the public repository https://github.com/Rbravo555/localECM.

Nonlinear model order reduction using local basis methods on RVEs: A parameter study and comparison of different variants

AbstractAll possible solutions to a static representative volume element (RVE) problem lie on a solution manifold, which is often quite low‐dimensional. Solutions of a hyperelastic 3D RVE problem, for example, are determined by the macroscopic deformation gradient, meaning that its solution manifold is only six‐dimensional. Established projection‐based model order reduction (MOR) methods such as the Proper Orthogonal Decomposition (POD), however, often require much higher‐dimensional approximation spaces to successfully capture such a nonlinear solution manifold on account of their linearity. One approach to modelling the solution manifold more closely is the local basis method by Amsallem et al.. It utilises a locally, rather than globally, linear approximation of the solution manifold. In this work, we investigate several variants of this algorithm with respect to their performance on an example RVE problem.

Solvability and Stability of Switched Discrete-time Singular Systems with the Same Switching Rules in Coefficient Matrices

The aim of this paper is to study the problem of solvability and stability for switched discrete-time linear singular (SDLS) systems with the same switching rules in coefficient matrices under Lipschitz perturbation. Firstly, we prove the unique existence of the solution, as well as describe the solution manifold. Secondly, by utilizing a Lyapunov function, we derive certain conditions that guarantee the stability of these systems. Finally, we illustrate our results with an example.

On the Solution Manifolds for Algebraic-Delay Systems

On the Solution Manifolds for Algebraic-Delay Systems

Model order reduction for the 1D Boltzmann-BGK equation: identifying intrinsic variables using neural networks

Kinetic equations are crucial for modeling non-equilibrium phenomena, but their computational complexity is a challenge. This paper presents a data-driven approach using reduced order models (ROM) to efficiently model non-equilibrium flows in kinetic equations by comparing two ROM approaches: proper orthogonal decomposition (POD) and autoencoder neural networks (AE). While AE initially demonstrate higher accuracy, POD’s precision improves as more modes are considered. Notably, our work recognizes that the classical POD model order reduction approach, although capable of accurately representing the non-linear solution manifold of the kinetic equation, may not provide a parsimonious model of the data due to the inherently non-linear nature of the data manifold. We demonstrate how AEs are used in finding the intrinsic dimension of a system and to allow correlating the intrinsic quantities with macroscopic quantities that have a physical interpretation.

Arbitrary-Lagrangian-Eulerian finite volume IMEX schemes for the incompressible Navier-Stokes equations on evolving Chimera meshes

In this article we design a finite volume semi-implicit IMEX scheme for the incompressible Navier-Stokes equations on evolving Chimera meshes. We employ a time discretization technique that separates explicit and implicit terms, accommodating the multi-scale nature of the governing equations, which involve both time scales of diffusion and advection operators. The finite volume approach for both explicit and implicit terms allows to encode into the nonlinear flux the velocity of displacement of the Chimera mesh via integration on moving cells. The numerical solution is then projected onto the physically meaningful solution manifold of non-solenoidal fields that stems from the energy equation. To attain second-order time accuracy, we employ semi-implicit IMEX Runge-Kutta schemes. These novel schemes are combined with a fractional-step method, thus the governing equations are eventually solved using a projection method to satisfy the divergence-free constraint of the velocity field. The implicit discretization of the viscous terms allows the CFL-type stability condition for the maximum admissible time step to be only defined by the relative fluid velocity referred to the movement of the frame and not depending on the viscous terms. Communication between different grid blocks is enabled through compact exchange of information from the fringe cells of one mesh block to the field cells of the other block. The continuity of the solution is recovered in one-shot during the solution of the arising algebraic systems by not involving neither direct discretization of the differential operators on fringe cells nor an iterative Schwartz-type method. Free-stream preservation property, i.e. compliance with the Geometric Conservation Law (GCL), is respected at the order of the scheme. The accuracy and capabilities of the new numerical schemes are proved through an extensive range of test cases, demonstrating ability to solve relevant benchmarks in the field of incompressible fluids.

On the solution manifolds for algebraic-delay systems

UDC 517.9 Differential equations with state-dependent delays specify a semiflow of continuously differentiable solution operators, in general, only on an associated submanifold of the Banach space C 1 ( [ - h ,0 ] , R n ) . We extend a recent result on the simplicity of these {\it solution manifolds} to systems in which the delay is given by the state only implicitly in an extra equation. These algebraic delay systems appear in various applications.

The best approximation problems between the least-squares solution manifolds of two matrix equations

In this paper, we will deal with the following two classes of best approximation problems about the linear manifolds: <b>Problem 1.</b> Given matrices $ A_1, B_1, C_1, $ and $ D_1 \in {\mathbb R}^{ m \times n} $, find $ d(L_1, L_2) = \min_{X\in L_1, Y\in L_2}\|X-Y\|, $ and find $ \hat{X}\in L_1, \hat{Y}\in L_2 $ such that $ \|\hat{X}-\hat{Y}\| = d(L_1, L_2) $, where $ L_1 = \left\{{X \in {\mathbb {SR}} ^{n \times n} \left|{\ \|A_1X-B_1\| = \min}\right.} \right\} $ and $ L_2 = \left\{{Y \in {\mathbb {SR}} ^{n \times n} \left|{\ \|C_1Y-D_1\| = \min}\right.} \right\} $. <b>Problem 2.</b> Given matrices $ A_2, B_2, E_2, F_2 \in {\mathbb R}^{ m \times n} $ and $ C_2, D_2, G_2, H_2 \in {\mathbb R}^{ n \times p} $, find $ d(L_3, L_4) = \min_{X\in L_3, Y\in L_4}\|X-Y\|, $ and find $ \tilde{X}\in L_3, \tilde{Y}\in L_4 $ such that $ \|\tilde{X}-\tilde{Y}\| = d(L_3, L_4) $, where $ L_3 = \left\{{X \in {\mathbb {R}}^{n \times n} \left|{\ \|A_2X-B_2\|^2+||XC_2-D_2\|^2 = \min}\right.} \right\} $ and $ L_4 = \left\{{Y \in {\mathbb {R}} ^{n \times n} \left|{\ \|E_2Y-F_2\|^2+||YG_2-H_2\|^2 = \min}\right.} \right\} $. We obtain explicit formulas for $ d(L_1, L_2) $ and $ d(L_3, L_4), $ and all the matrices in question by using the singular value decompositions and the canonical correlation decompositions of matrices.

A genetic-continuation algorithm for bifurcation analysis of nonlinear rotordynamic systems

This study presents a combined numerical scheme with genetic algorithm and continuation method for analysing nonlinear vibrations in rotor-bearing systems. A modified genetic algorithm is developed for solving multiple coexisting roots in nonlinear rotordynamic problem. Here, the genetic objective is the completion of periodicity for the boundary values of journal/pad phase states during the consecutive generations. The arc-length continuation takes the role to extend the solutions with respect to operating parameters such as the shaft spin speed and disc mass eccentricity. The resulting bifurcation diagrams, then, can comprehensively show the emergences and disappearances of multiple response states and their stability conditions in detail, which is unlike with the commonly applied numerical integration method. A finite model of heavily loaded Jeffcott rotor supported by six-pad tilting pad journal bearing is utilized to evaluate the developed algorithm. The 150 initial states for the journal and pad are generated, and the 8 optimal results are selected and inbreeded for evolution; it repeats for 15 generations. As results, multiple coexisting responses are identified at the same condition, and various bifurcation events such as periodic doubling, saddle-node are discovered from the solution manifolds.

Reinforcement-based processes actively regulate motor exploration along redundant solution manifolds.

From a baby's babbling to a songbird practising a new tune, exploration is critical to motor learning. A hallmark of exploration is the emergence of random walk behaviour along solution manifolds, where successive motor actions are not independent but rather become serially dependent. Such exploratory random walk behaviour is ubiquitous across species' neural firing, gait patterns and reaching behaviour. The past work has suggested that exploratory random walk behaviour arises from an accumulation of movement variability and a lack of error-based corrections. Here, we test a fundamentally different idea-that reinforcement-based processes regulate random walk behaviour to promote continual motor exploration to maximize success. Across three human reaching experiments, we manipulated the size of both the visually displayed target and an unseen reward zone, as well as the probability of reinforcement feedback. Our empirical and modelling results parsimoniously support the notion that exploratory random walk behaviour emerges by utilizing knowledge of movement variability to update intended reach aim towards recently reinforced motor actions. This mechanism leads to active and continuous exploration of the solution manifold, currently thought by prominent theories to arise passively. The ability to continually explore muscle, joint and task redundant solution manifolds is beneficial while acting in uncertain environments, during motor development or when recovering from a neurological disorder to discover and learn new motor actions.

To the Theory of Decaying Turbulence

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension d>2. This solution family is equivalent to a fractal curve in complex space Cd with random steps parametrized by N Ising variables σi=±1, in addition to a rational number pq and an integer winding number r, related by ∑σi=qr. This equivalence provides a dual theory describing a strong turbulent phase of the Navier-Stokes flow in Rd space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. From a mathematical point of view, this theory implements a stochastic solution of the unforced Navier-Stokes equations. For a theoretical physicist, this is a quantum statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size N→∞ or its chemical potential μ→0. The system with fixed N has different asymptotics at odd and even N→∞, but the limit μ→0 is well defined. The energy dissipation rate is analytically calculated as a function of μ using methods of number theory. It grows as ν/μ2 in the continuum limit μ→0, leading to anomalous dissipation at μ∝ν→0. The same method is used to compute all the local vorticity distribution, which has no continuum limit but is renormalizable in the sense that infinities can be absorbed into the redefinition of the parameters. The small perturbation of the fixed manifold satisfies the linear equation we solved in a general form. This perturbation decays as t−λ, with a continuous spectrum of indexes λ in the local limit μ→0. The spectrum is determined by a resolvent, which is represented as an infinite product of 3⊗3 matrices depending of the element of the Euler ensemble.

On Stabilizability of Nonbilinear Perturbed Descriptor Systems

One way in which nonlinear descriptor systems of (index-k) naturally arise is through semiexplicit differential-algebraic equations. The study considers the nonbilinear dynamical systems which are described by the class of higher-index differential-algebraic equations (DAEs). Their nature is analysed both quantitatively and qualitatively, and stability characteristics are presented for their solution. Higher-index differential-algebraic systems seem to show inherent shaky around their solution manifolds. The often use of logarithmic norms is for the estimation of stability and perturbation bounds in linear ordinary differential equations (ODEs). The question of how to apply the notation of logarithmic norms to nonlinear DAEs has long been an open question. Other problem extensions including nonlinear dynamics and nonbilinear DAEs need subtle modification of the logarithmic norms. The logarithmic norm is combined by conceptual focus with the finite-time stability criterion in order to treat nonbilinear DAEs with the aim of covering some unbounded operators. This means we obtain the perturbation bounds from differential inequalities for a norm by the use of the relationship between Dini derivatives and semi-inner products. A numerical result obtained when tested on the nonbilinear mechanical system with a larger scale showed that the method was highly efficient and accurate and particularly suitable for nonbilinear DAEs.

A Newton’s iteration converges quadratically to nonisolated solutions too

The textbook Newton’s iteration is practically inapplicable on nonisolated solutions of unregularized nonlinear systems. With a simple modification, a version of Newton’s iteration regains its local quadratic convergence to nonisolated zeros of smooth mappings assuming the solutions are semiregular as properly defined regardless of whether the system is square, underdetermined or overdetermined. Furthermore, the iteration serves as a de facto regularization mechanism for computing singular zeros from empirical data. Even if the given system is perturbed so that the nonisolated solution disappears, the iteration still locally converges to a stationary point that approximates a solution of the underlying system with an error bound in the same order of the data accuracy. Geometrically, the iteration approximately converges to the nearest point on the solution manifold. This extension simplifies nonlinear system modeling by eliminating the zero isolation process and enables a wide range of applications in algebraic computation.

A new family of semi-implicit Finite Volume/Virtual Element methods for incompressible flows on unstructured meshes

We introduce a new family of high order accurate semi-implicit schemes for the solution of nonlinear time-dependent systems of partial differential equations (PDE) on unstructured polygonal meshes. The time discretization is based on a splitting between explicit and implicit terms that may arise either from the multi-scale nature of the governing equations, which involve both slow and fast scales, or in the context of projection methods, where the numerical solution is projected onto the physically meaningful solution manifold. We propose to use a high order finite volume (FV) scheme for the explicit terms, hence ensuring conservation property and robustness across shock waves, while the virtual element method (VEM) is employed to deal with the discretization of the implicit terms, which typically requires an elliptic problem to be solved. The numerical solution is then transferred via suitable L2 projection operators from the FV to the VEM solution space and vice-versa. High order time accuracy is then achieved using the semi-implicit IMEX Runge–Kutta schemes, and the novel schemes are proven to be asymptotic preserving (AP) and well-balanced (WB). As representative models, we choose the shallow water equations (SWE), thus handling multiple time scales characterized by a different Froude number, and the incompressible Navier–Stokes equations (INS), which are solved at the aid of a projection method to satisfy the solenoidal constraint of the velocity field. Furthermore, an implicit discretization for the viscous terms is also devised for the INS model, which is based on the VEM technique. Consequently, the CFL-type stability condition on the maximum admissible time step is based only on the fluid velocity and not on the celerity nor on the viscous eigenvalues. A large suite of test cases demonstrates the accuracy and the capabilities of the new family of schemes to solve relevant benchmarks in the field of incompressible fluids.