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Meta-model or surrogate model is a model of a model, and meta-modelling is the process of generating such meta-models. Metamodelling or meta-modelling is the analysis, construction and development of the frames, rules, constraints, models and theories applicable and useful for modelling a predefined class of problems. Meta-model can be a mathematical relation or algorithm representing input and output relations. A model is an abstraction of phenomena in the real world; a meta-model is yet another abstraction, highlighting properties of the model itself. Various types of metamodels include polynomia
l equations, neural network, Kriging, etc. Metamodeling typically involves studying the output and input relationships and then fitting right metamodels to represent that behaviour. Decision analytic models are frequently required for conducting cost-effectiveness analyses, and a probabilistic sensitivity analysis (PSA) will have to be performed on any model to assess model output uncertainty induced by uncertainty in the model inputs.
Figure shows the response curve of surrogate model Figure shows the procedure followed to create a surrogate model This figure explains the surrogate modelling for a particular engineering
and actual model of a random sample model for a problem from the training data set problem, specially the computation cost of the analysis is too high.
The problem arises when the model is computationally expensive to run, for example CFD analysis or weather forecasting. A model developer may be deterred from building a sufficiently realistic model due to the consequent computational problems. Complex mathematical models are used in many fields of science and technology, and it is common to investigate the consequences of input uncertainty using approaches such as PSA. An established technique for dealing with computationally expensive models is to construct a meta-model (also known as an emulator). A meta-model is a statistical representation of the original model that can be used to obtain very fast approximations of the model output. A meta-model can be constructed using a relatively small number of runs of the original model, and so be can used to conduct a PSA with accuracy comparable to Monte Carlo, but with far fewer model runs. Meta-modelling involves the use of statistical regression to learn the model input-output relationship. A popular regression method used in meta-modelling involves the use of Gaussian processes. Gaussian process regression is a non-parametric method, and so it is not necessary to make restrictive assumptions about the input-output relationship in the decision model.
A surrogate model is an engineering method used when an outcome of interest cannot be easily directly measured, so a model of the outcome is used instead. Most engineering design problems require experiments and/or simulations to evaluate design objective and constraint functions as function of design variables. For example, in order to find the optimal airfoil shape for an aircraft wing, an engineer simulates the air flow around the wing for different shape variables (length, curvature, material, ..). For many real world problems, however, a single simulation can take many minutes, hours, or even days to complete. As a result, routine tasks such as design optimization, design space exploration, sensitivity analysis and what-if analysis become impossible since they require thousands or even millions of simulation evaluations. One way of alleviating this burden is by constructing approximation models, known as surrogate models, response surface models, meta-models or emulators, that mimic the behaviour of the simulation model as closely as possible while being computationally cheaper to evaluate. Surrogate models are constructed using a data-driven, bottom-up approach. The exact, inner working of the simulation code is not assumed to be known (or even understood), solely the input-output behaviour is important. A model is constructed based on modelling the response of the simulator to a limited number of intelligently chosen data points. This approach is also known as behavioral modeling or black-box modeling, though the terminology is not always consistent. When only a single design variable is involved, the process is known as curve fitting.
An example of curve fitting. A surrogate model to imitate 1 parameter model.
The scientific challenge of surrogate modelling is the generation of a surrogate that is as accurate as possible, using as few simulation evaluations as possible. The process comprises three major steps which may be interleaved iteratively:
Sample selection (also known as sequential design, optimal experimental design (OED) or active learning)
Construction of the surrogate model and optimizing the model parameters (bias–variance trade-off)
Appraisal of the accuracy of the surrogate.
The accuracy of the surrogate depends on the number and location of samples (expensive experiments or simulations) in the design space. Various design of experiments (DOE) techniques cater to different sources of errors, in particular errors due to noise in the data or errors due to an improper surrogate model.
Types of surrogate models
The most popular surrogate models are polynomial response surfaces, Kriging, support vector machines, space mapping and artificial neural networks. For some problems, the nature of true function is not known a priori so it is not clear which surrogate model will be most accurate. In addition, there is no consensus on how to obtain the most reliable estimates of the accuracy of a given surrogate. Many other problems have known physics properties. In these cases, physics-based surrogates such as space-mapping-based models are the most efficient.
An important distinction can be made between two different applications of surrogate models: design optimization and design space approximation (also known as emulation). In design space approximation, one is not interested in finding the optimal parameter vector but rather in the global behaviour of the system. Here the surrogate is tuned to mimic the underlying model as closely as needed over the complete design space. Such surrogates are a useful, cheap way to gain insight into the global behaviour of the system. Optimization can still occur as a post processing step, although with no update procedure the optimum found cannot be validated.
---to be continued
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