The control of dynamical systems by neural networks

In this paper the use of neural networks for the control of dynamical systems is considered. Both identification and feedback control aspects are discussed as well as the types of system for which neural networks can provide a useful technique. Multi-layer Perceptron and Radial Basis function neural network types are looked at, with an emphasis on the latter. It is shown how basis function centre selection is a critical part of the implementation process and that multivariate clustering algorithms can be an extremely useful tool for finding centres.

Stochastic perturbations to dynamical systems: a response theory approach

Using the formalism of the Ruelle response theory, we study how the invariant measure of an Axiom A dynamical system changes as a result of adding noise, and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics. We first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the additive and in the multiplicative case. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. This provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings. Using Kramers-Kronig theory, it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case analysed. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide an example of application of our results by considering the spatially extended chaotic Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations. Taking into account the chaotic hypothesis, we expect that our results have practical relevance for a more general class of system than those belonging to Axiom A.

A general method for finding extremal states of Hamiltonian dynamical systems, with applications to perfect fluids

In addition to the Hamiltonian functional itself, non-canonical Hamiltonian dynamical systems generally possess integral invariants known as ‘Casimir functionals’. In the case of the Euler equations for a perfect fluid, the Casimir functionals correspond to the vortex topology, whose invariance derives from the particle-relabelling symmetry of the underlying Lagrangian equations of motion. In a recent paper, Vallis, Carnevale & Young (1989) have presented algorithms for finding steady states of the Euler equations that represent extrema of energy subject to given vortex topology, and are therefore stable. The purpose of this note is to point out a very general method for modifying any Hamiltonian dynamical system into an algorithm that is analogous to those of Vallis etal. in that it will systematically increase or decrease the energy of the system while preserving all of the Casimir invariants. By incorporating momentum into the extremization procedure, the algorithm is able to find steadily-translating as well as steady stable states. The method is applied to a variety of perfect-fluid systems, including Euler flow as well as compressible and incompressible stratified flow.

A dynamical systems framework for intermittent data assimilation

We consider the problem of discrete time filtering (intermittent data assimilation) for differential equation models and discuss methods for its numerical approximation. The focus is on methods based on ensemble/particle techniques and on the ensemble Kalman filter technique in particular. We summarize as well as extend recent work on continuous ensemble Kalman filter formulations, which provide a concise dynamical systems formulation of the combined dynamics-assimilation problem. Possible extensions to fully nonlinear ensemble/particle based filters are also outlined using the framework of optimal transportation theory.

Towards a general theory of extremes for observables of chaotic dynamical systems

In this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the cho- sen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan–Yorke dimension of the attractor. Preliminary numer- ical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.

A joint state and parameter estimation scheme for nonlinear dynamical systems

We present a novel algorithm for concurrent model state and parameter estimation in nonlinear dynamical systems. The new scheme uses ideas from three dimensional variational data assimilation (3D-Var) and the extended Kalman filter (EKF) together with the technique of state augmentation to estimate uncertain model parameters alongside the model state variables in a sequential filtering system. The method is relatively simple to implement and computationally inexpensive to run for large systems with relatively few parameters. We demonstrate the efficacy of the method via a series of identical twin experiments with three simple dynamical system models. The scheme is able to recover the parameter values to a good level of accuracy, even when observational data are noisy. We expect this new technique to be easily transferable to much larger models.

Assessing the performance of dynamical trajectory estimates

Estimating trajectories and parameters of dynamical systems from observations is a problem frequently encountered in various branches of science; geophysicists for example refer to this problem as data assimilation. Unlike as in estimation problems with exchangeable observations, in data assimilation the observations cannot easily be divided into separate sets for estimation and validation; this creates serious problems, since simply using the same observations for estimation and validation might result in overly optimistic performance assessments. To circumvent this problem, a result is presented which allows us to estimate this optimism, thus allowing for a more realistic performance assessment in data assimilation. The presented approach becomes particularly simple for data assimilation methods employing a linear error feedback (such as synchronization schemes, nudging, incremental 3DVAR and 4DVar, and various Kalman filter approaches). Numerical examples considering a high gain observer confirm the theory.

Tropospheric planetary wave dynamics and mixture modeling: Two preferred regimes and a regime shift

Investigation of preferred structures of planetary wave dynamics is addressed using multivariate Gaussian mixture models. The number of components in the mixture is obtained using order statistics of the mixing proportions, hence avoiding previous difficulties related to sample sizes and independence issues. The method is first applied to a few low-order stochastic dynamical systems and data from a general circulation model. The method is next applied to winter daily 500-hPa heights from 1949 to 2003 over the Northern Hemisphere. A spatial clustering algorithm is first applied to the leading two principal components (PCs) and shows significant clustering. The clustering is particularly robust for the first half of the record and less for the second half. The mixture model is then used to identify the clusters. Two highly significant extratropical planetary-scale preferred structures are obtained within the first two to four EOF state space. The first pattern shows a Pacific-North American (PNA) pattern and a negative North Atlantic Oscillation (NAO), and the second pattern is nearly opposite to the first one. It is also observed that some subspaces show multivariate Gaussianity, compatible with linearity, whereas others show multivariate non-Gaussianity. The same analysis is also applied to two subperiods, before and after 1978, and shows a similar regime behavior, with a slight stronger support for the first subperiod. In addition a significant regime shift is also observed between the two periods as well as a change in the shape of the distribution. The patterns associated with the regime shifts reflect essentially a PNA pattern and an NAO pattern consistent with the observed global warming effect on climate and the observed shift in sea surface temperature around the mid-1970s.

Non-commutative symbolic coding

We give a non-commutative generalization of classical symbolic coding in the presence of a synchronizing word. This is done by a scattering theoretical approach. Classically, the existence of a synchronizing word turns out to be equivalent to asymptotic completeness of the corresponding Markov process. A criterion for asymptotic completeness in general is provided by the regularity of an associated extended transition operator. Commutative and non-commutative examples are analysed.

Very large inverse problems in atmosphere and ocean modelling

For the very large nonlinear dynamical systems that arise in a wide range of physical, biological and environmental problems, the data needed to initialize a numerical forecasting model are seldom available. To generate accurate estimates of the expected states of the system, both current and future, the technique of ‘data assimilation’ is used to combine the numerical model predictions with observations of the system measured over time. Assimilation of data is an inverse problem that for very large-scale systems is generally ill-posed. In four-dimensional variational assimilation schemes, the dynamical model equations provide constraints that act to spread information into data sparse regions, enabling the state of the system to be reconstructed accurately. The mechanism for this is not well understood. Singular value decomposition techniques are applied here to the observability matrix of the system in order to analyse the critical features in this process. Simplified models are used to demonstrate how information is propagated from observed regions into unobserved areas. The impact of the size of the observational noise and the temporal position of the observations is examined. The best signal-to-noise ratio needed to extract the most information from the observations is estimated using Tikhonov regularization theory. Copyright © 2005 John Wiley & Sons, Ltd.

Unifying syntactic theory and sentence processing difficulty through a connectionist minimalist parser

Syntactic theory provides a rich array of representational assumptions about linguistic knowledge and processes. Such detailed and independently motivated constraints on grammatical knowledge ought to play a role in sentence comprehension. However most grammar-based explanations of processing difficulty in the literature have attempted to use grammatical representations and processes per se to explain processing difficulty. They did not take into account that the description of higher cognition in mind and brain encompasses two levels: on the one hand, at the macrolevel, symbolic computation is performed, and on the other hand, at the microlevel, computation is achieved through processes within a dynamical system. One critical question is therefore how linguistic theory and dynamical systems can be unified to provide an explanation for processing effects. Here, we present such a unification for a particular account to syntactic theory: namely a parser for Stabler's Minimalist Grammars, in the framework of Smolensky's Integrated Connectionist/Symbolic architectures. In simulations we demonstrate that the connectionist minimalist parser produces predictions which mirror global empirical findings from psycholinguistic research.

Stability criteria for the contextual emergence of macrostates in neural networks

More than thirty years ago, Amari and colleagues proposed a statistical framework for identifying structurally stable macrostates of neural networks from observations of their microstates. We compare their stochastic stability criterion with a deterministic stability criterion based on the ergodic theory of dynamical systems, recently proposed for the scheme of contextual emergence and applied to particular inter-level relations in neuroscience. Stochastic and deterministic stability criteria for macrostates rely on macro-level contexts, which make them sensitive to differences between different macro-levels.

A neurofuzzy network knowledge extraction and extended gram-Schmidt algorithm for model subspace decomposition

This paper introduces a new neurofuzzy model construction and parameter estimation algorithm from observed finite data sets, based on a Takagi and Sugeno (T-S) inference mechanism and a new extended Gram-Schmidt orthogonal decomposition algorithm, for the modeling of a priori unknown dynamical systems in the form of a set of fuzzy rules. The first contribution of the paper is the introduction of a one to one mapping between a fuzzy rule-base and a model matrix feature subspace using the T-S inference mechanism. This link enables the numerical properties associated with a rule-based matrix subspace, the relationships amongst these matrix subspaces, and the correlation between the output vector and a rule-base matrix subspace, to be investigated and extracted as rule-based knowledge to enhance model transparency. The matrix subspace spanned by a fuzzy rule is initially derived as the input regression matrix multiplied by a weighting matrix that consists of the corresponding fuzzy membership functions over the training data set. Model transparency is explored by the derivation of an equivalence between an A-optimality experimental design criterion of the weighting matrix and the average model output sensitivity to the fuzzy rule, so that rule-bases can be effectively measured by their identifiability via the A-optimality experimental design criterion. The A-optimality experimental design criterion of the weighting matrices of fuzzy rules is used to construct an initial model rule-base. An extended Gram-Schmidt algorithm is then developed to estimate the parameter vector for each rule. This new algorithm decomposes the model rule-bases via an orthogonal subspace decomposition approach, so as to enhance model transparency with the capability of interpreting the derived rule-base energy level. This new approach is computationally simpler than the conventional Gram-Schmidt algorithm for resolving high dimensional regression problems, whereby it is computationally desirable to decompose complex models into a few submodels rather than a single model with large number of input variables and the associated curse of dimensionality problem. Numerical examples are included to demonstrate the effectiveness of the proposed new algorithm.