Approximation Of Nonlinear Dynamical Systems Research Articles

Tensor-Based Data-Driven Identification of Partial Differential Equations

Abstract We present a tensor-based method for model selection which identifies the unknown partial differential equation that governs a dynamical system using only spatiotemporal measurements. The method circumvents a disadvantage of standard matrix-based methods which typically have large storage consumption. Using a recently developed multidimensional approximation of nonlinear dynamical systems (MANDy), we collect the nonlinear and partial derivative terms of the measured data and construct a low-rank dictionary tensor in the tensor-train (TT) format. A tensor-based linear regression problem is then built, which balances the learning accuracy, model complexity, and computational efficiency. An algebraic expression of the unknown equations can be extracted. Numerical results are demonstrated on datasets generated by the wave equation, the Burgers' equation, and a few parametric partial differential equations (PDEs).

Multidimensional Approximation of Nonlinear Dynamical Systems

A key task in the field of modeling and analyzing nonlinear dynamical systems is the recovery of unknown governing equations from measurement data only. There is a wide range of application areas for this important instance of system identification, ranging from industrial engineering and acoustic signal processing to stock market models. In order to find appropriate representations of underlying dynamical systems, various data-driven methods have been proposed by different communities. However, if the given data sets are high-dimensional, then these methods typically suffer from the curse of dimensionality. To significantly reduce the computational costs and storage consumption, we propose the method multidimensional approximation of nonlinear dynamical systems (MANDy) which combines data-driven methods with tensor network decompositions. The efficiency of the introduced approach will be illustrated with the aid of several high-dimensional nonlinear dynamical systems.

Linearly Recurrent Autoencoder Networks for Learning Dynamics

This paper describes a method for learning low-dimensional approximations of nonlinear dynamical systems, based on neural network approximations of the underlying Koopman operator. Extended Dynamic...

Numerical Approximations of the Dynamical System Generated by Burgers’ Equation with Neumann–Dirichlet Boundary Conditions

Using Burgers' equation with mixed Neumann-Dirichlet boundary conditions, we highlight a problem that can arise in the numerical approximation of nonlinear dynamical systems on computers with a finite precision floating point number system. We describe the dynamical system generated by Burgers' equation with mixed boundary conditions, summarize some of its properties and analyze the equilibrium states for finite dimensional dynamical systems that are generated by numerical approxima- tions of this system. It is important to note that there are two fundamental differences between Burgers' equation with mixed Neumann-Dirichlet boundary conditions and Burgers' equation with both Dirich- let boundary conditions. First, Burgers' equation with homogenous mixed boundary conditions on a finite interval cannot be linearized by the Cole-Hopf transformation. Thus, on finite intervals Burgers' equation with a homogenous Neumann boundary condition is truly nonlinear. Second, the nonlinear term in Burgers' equation with a homogenous Neumann boundary condition is not conservative. This structure plays a key role in understanding the complex dynamics generated by Burgers' equation with a Neumann boundary condition and how this structure impacts numerical approximations. The key point is that, regardless of the particular numerical scheme, finite precision arithmetic will always lead to numerically generated equilibrium states that do not correspond to equilibrium states of the Burgers' equation. In this paper we establish the existence and stability properties of these numerical stationary solutions and employ a bifurcation analysis to provide a detailed mathematical explanation of why numerical schemes fail to capture the correct asymptotic dynamics. We extend the results in (E. Allen, J.A. Burns, D.S. Gilliam, J. Hill and V.I. Shubov, Math. Comput. Modelling 35 (2002) 1165-1195) and prove that the effect of finite precision arithmetic persists in generating a nonzero numerical false solution to the stationary Burgers' problem. Thus, we show that the results obtained in (E. Allen, J.A. Burns, D.S. Gilliam, J. Hill and V.I. Shubov, Math. Comput. Modelling 35 (2002) 1165-1195) are not dependent on a specific time marching scheme, but are generic to all convergent numerical approximations of Burgers' equation.

AFAS - adaptive fast approximate simulation for non-linear model reduction

In this work, we examine the problem of selecting linearisation points for trajectory piecewise linear (TPWL) approximation of non-linear dynamical systems. Linearisation point selection is a crucial step in the TPWL process, the quality and complexity of the approximation rests on it. In contrast to the popular approaches wherein linearisations are done at constant, pre-selected Euclidean distances in the state-space, we propose a new and simple error measure that helps in assessing the linearisation point requirement at different points on the non-linear system trajectory. Based on this error measure a new scheme to simulate the non-linear system, create linearisations at viable points and obtain a better TPWL approximation is presented. Finally, we substantiate our observations and propositions by detailed numerical tests on two non-linear circuits.

Towards Accurate PWL Approximations of Parameter-Dependent Nonlinear Dynamical Systems With Equilibria and Limit Cycles

This paper deals with piecewise-linear (PWL) approximations of nonlinear dynamical systems dependent on parameters and allowing the presence of few equilibria and/or limit cycles only. A method to derive the parameters of the PWL model is proposed that is based on the minimization of functionals defined to take into account a priori some dynamical features of the systems to be approximated. The method is validated by applying it to two simple dynamical systems, i.e., the cusp bifurcation normal form and the supercritical Hopf bifurcation normal form. The robustness of the approximations is checked, with a view to circuit implementations

PWL approximation of nonlinear dynamical systems, part I: structural stability

This paper and its companion address the problem of the approximation/identification of nonlinear dynamical systems depending on parameters, with a view to their circuit implementation. The proposed method is based on a piecewise-linear approximation technique. In particular, this paper describes the approximation method and applies it to some particularly significant dynamical systems (topological normal forms). The structural stability of the PWL approximations of such systems is investigated through a bifurcation analysis (via continuation methods).

PWL approximation of nonlinear dynamical systems, part II: identification issues

This paper and its companion address the problem of the approximation/identification of nonlinear dynamical systems depending on parameters, with a view to their circuit implementation. The proposed method is based on a piecewise-linear approximation technique. In particular, this paper describes a black-box identification method based on state space reconstruction and PWL approximation, and applies it to some particularly significant dynamical systems (two topological normal forms and the Colpitts oscillator).

Piecewise-Linear Approximation of Nonlinear Dynamical Systems

The piecewise-linear (PWL) approximation technique developed by Julia/spl acute/n et al. in the past few years is applied to find approximate models of dynamical systems dependent on given numbers of state variables and parameters. Referring to some significant examples, i.e., topological normal forms, it is shown that a PWL dynamical system approximating a given smooth system can preserve its main features. In particular, if the approximation accuracy increases, the equivalence between approximating and approximated systems shifts from qualitative to quantitative. The validity of the proposed approach is eventually tested by use of a severe nonlinear example, i.e., the Rosenzweig-MacArthur system, which describes the population dynamics in a tritrophic food chain model.

Neurocontrol of nonlinear systems via local memory neurons

In this report, we consider the part of our work which concerns the approximation of nonlinear dynamic systems using neural networks. Based on a new paradigm of neurons with local memory (NNLM), we discuss the representation of control systems by neural networks. Using this formulation, the basic issues of controllability and observability for the dynamic system are addressed. A separation principle of learning and control is presented for NNLM, showing that the weights of the network do not affect its dynamics. Theoretical issues concerning local linearization via a coordinate transformation and nonlinear feedback are discussed. For illustration of the approach simulation results for nonlinear control of an aircraft encountering wind shear on take-off is presented.

Category of topological jet manifolds and certain applications in the theory of nonlinear infinite-dimensional dynamical systems

A new category of topological jet manifolds is proposed for the purpose of investigating exact finite-dimensional approximations of nonlinear dynamical systems on infinite-dimensional functional manifolds. Differential geometry structures on these manifolds and their applications to the theory of integrability in quadratures of nonlinear dynamical Lax-type systems are studied.

Approximation of nonlinear dynamic systems by rational series

Given an analytic system, we compute a bilinear system of minimal dimension which approximates it up to order k (i.e. the outputs of these two systems have the same Taylor expansion up to order k). The algorithm is based on noncommutative series computation: let s be the generating series of the analytic system; then a rational series g is constructed, whose coefficients are equal to those of s, for all words of length smaller than or equal to k. These words are digitally encoded, in order to simplify the computations of the Hankel matrices of s and g. We then associate with g, a bilinear system, which is a solution to our problem. Another method may be used for computing a bilinear system which approximates a given analytic system ( S). We associate with ( S) an R-automaton of vector fields and build the truncated automaton by cancelling all the states which have the following property: the length of the shortest successful path labelled by a word that gets through this state is strictly greater than k. Then, the number of states of this truncated automaton yields the dimension (not necessarily minimal) of the state-space.