Improved modeling by coupling imperfect models

doi:10.1016/j.cnsns.2011.11.003

Communications in Nonlinear Science and Numerical Simulation

Volume 17, Issue 7, July 2012, Pages 2741-2751

https://doi.org/10.1016/j.cnsns.2011.11.003 Get rights and content

Abstract

Most of the existing approaches for combining models representing a single real-world phenomenon into a multi-model ensemble combine the models a posteriori. Alternatively, in our method the models are coupled into a supermodel and continuously communicate during learning and prediction. The method learns a set of coupling coefficients from short past data in order to unite the different strengths of the models into a better representation of the observed phenomenon. The method is examined using the Lorenz oscillator, which is altered by introducing parameter and structural differences for creating imperfect models. The short past data is obtained by the standard oscillator, and different weight is assigned to each sample of the past data. The coupling coefficients are learned by using a quasi-Newton method and an evolutionary algorithm. We also introduce a way for reducing the supermodel, which is particularly useful for models of high complexity. The results reveal that the proposed supermodel gives a very good representation of the truth even for substantially imperfect models and short past data, which suggests that the super-modeling is promising in modeling real-world phenomena.

Highlights

► We couple an ensemble of existing models representing a single real-world phenomena. ► Models interactively exchange information during learning and prediction. ► Coupling coefficients are learned from short past data of the observed phenomenon. ► Examination with Lorenz systems with radical imperfections show good approximation. ► The ensemble is reduced to be made useful for models of high complexity.

Introduction

There are three main aspects of modeling real-world phenomena. First, in the structure identification problem one deals with establishing appropriate equations (structure) governing the evolution in time of the variables describing the modeled phenomenon. Second, in the parameter estimation problem one determines the acceptably accurate values for the parameters of the modeled real system. These two general problems are addressed in [1], and a more specific overview regarding nonlinear dynamics and chaos is given in [2]. Third aspect of modeling, not discussed in this paper, is data assimilation [3], which means estimation of the model’s initial conditions based on limited observable data.

Complete modeling of a real-world phenomenon is desirable, but it is difficult to be accomplished due to several reasons: limited computing power, limited knowledge of the process (system) modeled, limited observable data of the modeled process, and so on. As a step toward this goal, we consider an approach based on building an ensemble of dynamic models. These models often represent distinct aspects of a same process (system) with different success. Therefore, the goal is to provide a method for coupling imperfect models to each other in a way that they better approximate the modeled process, called here the truth, thus creating an interactive ensemble of (imperfect) models, called here a supermodel. There are two aspects in creating the ensemble of models: structure identification problem in which we establish the structure of the coupling mechanism, and parameter estimation problem in which the parameters of the coupling equations are estimated. The coupling mechanism could be described with linear equations, nonlinear equations, and even with a set of differential equations, while the coupling coefficients could be estimated using various optimization methods.

Numerous approaches for combining multiple models for representing a single real process have already been explored [4], particularly for future climate projections [5], [6]. However, most of these methods have been designed to aggregate the predictions obtained with the different models. Alternatively, in the super-modeling approach the multiple models are coupled so that they can interactively exchange information continuously during the learning and prediction processes. This approach has been successfully used in [7], where an ensemble of atmospheric models is coupled to one oceanographic model, which in turn gives better results than if just one atmospheric model is used. The basic concept and justifiability of the super-modeling approach are already presented in [8], where three imperfect models with parameter differences are coupled into one supermodel to represent the Lorenz 63, Lorenz 84 and Rossler systems. The authors use linear equations to couple the imperfect models and assume that there are data available from which the coupling coefficients can be learned. Hence, the problem is reduced to learning the coupling coefficients from the available data.

In this paper we also primarily consider linear equations that describe coupling between imperfect models and focus on parameter estimation problem in the ensemble of imperfect models. We assume that we have data that are generated by a chaotic system (long-term unpredictable deterministic system) from which one can learn the coupling coefficients, but unlike in [8] we assume significantly smaller amount of data available. We consider imperfect models that differ from the truth model in two ways: first, the parameters of the imperfect models are perturbed as in [8], and second, the structure of differential equations of the imperfect models is different than that of the truth model. We assume that we cannot improve the imperfect models by estimating their own parameters and/or identifying their own structure. The small amount of available data and the larger order of imperfections require modifications of the learning process applied in [8]. Therefore, we introduce a new weight function that is used during the learning and we explore different optimization methods for finding a suitable set of coupling coefficients. We also propose one possible approach for reducing the ensemble of imperfect models, which makes the supermodel applicable to models of greater complexity.

This is the outline of the paper. In Section 2 we describe the main concepts of our super-modeling approach: definition, learning and validation of the supermodel. In Section 3 we examine the super-modeling approach by using the Lorenz oscillator, first by coupling models with only parameter differences and then models with both parameter and structural differences. We propose a way to reduce the ensemble of imperfect models in the end of Section 3, and Section 4 concludes the paper.

Section snippets

Definition of the supermodel

In the proposed method it is supposed that a number of M imperfect models representing a single real process already exist and the goal is to create an improved supermodel by coupling the existing models, as presented in [8]. It is assumed that each imperfect model μ is described by an N-dimensional system: ${\dot{x}}_{μ} = f_{μ} (x_{μ}, p_{μ}),$ where $x_{μ} = {[\begin{matrix} x_{μ}^{1} & x_{μ}^{2} & \dots & x_{μ}^{N} \end{matrix}]}^{T} \in R^{N}$ is a state variables vector, p_μ ∈ R^p is a vector of the model parameters, and f_μ : R^N×p → R^N is a parameterized dynamic of the model. In the presentation of

Supermodel composed of altered Lorenz oscillators

To present and test the method of coupling imperfect models we use the well known Lorenz oscillator [14], which exhibits a chaotic behavior suggestive of that of the atmosphere. As “truth” a Lorenz oscillator is used, as given in Eq. (5), with parameter values as originally presented by Lorenz (σ = 10, ρ = 28 and β = 8/3). The state variables vector is x = [x y z] and the parameters vector is p = [σ ρ β]. $\begin{matrix} \dot{x} = σ (y - x), \\ \dot{y} = x (ρ - z) - y, \\ \dot{z} = xy - β z . \end{matrix}$

On the other hand, for the ensemble of models we use three Lorenz

Conclusions

In this paper we proposed a modified super-modeling approach in which imperfect models of a real observable system are combined and continuously communicate during learning and prediction. The models interact through linear connections with coefficients that are learned from short past record of observed data by minimizing mean squared errors in short consecutive runs. The main result is that the supermodel outperforms the individual imperfect models in the case when models have both structural

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View full text

Improved modeling by coupling imperfect models

Abstract

Highlights

Introduction

Section snippets

Definition of the supermodel

Supermodel composed of altered Lorenz oscillators

Conclusions

System identification: theory for the user

Modeling nonlinear dynamics and chaos: a review

Math Probl Eng

Atmospheric modeling, data assimilation and predictability

Bayesian model averaging: a tutorial

Stat Sci

The use of the multi-model ensemble in probabilistic climate projections

Philos Trans Roy Soc A

Challenges in combining projections from multiple climate models

J Climate

Interactive coupled ensemble: a new coupling strategy for GCMs

Geophys Res Lett