Thread: Generic methods to emulate derivatives¶
Overview¶
This thread describes how we can build an emulator with which we can predict the derivatives of the model output with respect to the inputs. If we have derivative information available, either from an adjoint model or some other means, we can include that information when emulating derivatives. This is similar to the variant thread on emulators with derivative information (ThreadVariantWithDerivatives) which includes derivative information when emulating function output. If the adjoint to a simulator doesn’t exist and we don’t wish to obtain derivative information through another method, it is still possible to emulate model derivatives with just the function output.
The emulator¶
The derivatives of a posterior Gaussian process remain Gaussian processes with mean and covariance functions obtained by the relevant derivatives of the posterior mean and covariance functions. This can be applied to any Gaussian process emulator. The process of building an emulator of derivatives with the fully Bayesian approach is given in the procedure page ProcBuildEmulateDerivsGP. This covers building a Gaussian process emulator of derivatives with just function output, an extension of the core thread ThreadCoreGP, and a Gaussian process emulator of derivatives built with function output and derivative information, an extension of ThreadVariantWithDerivatives.
The result is a Gaussian process emulator of derivatives which will correctly represent any derivatives in the training data, but it is always important to validate the emulator against additional derivative information. For the core problem, the process of validation is described in the procedure page ProcValidateCoreGP. Although here we are interested in emulating derivatives, as we know the derivatives of a Gaussian process remain a Gaussian process, we can apply the same validation techniques as for the core problem. We require a validation design \(D^\prime\) which consists of points where we want to obtain validation derivatives. An adjoint is then run at these points; if an appropriate adjoint does not exist the derivatives are obtained through another technique, for example finite differences. If any local sensitivity analysis has already been performed on the simulator, some derivatives may already have been obtained and can be used here for validation. Then in the case of a linear mean function, weak prior information on hyperparameters \(\beta\) and \(\sigma\), and a single posterior estimate of \(\delta\), the predictive mean vector, \(m^*\), and the predictive covariance matrix, \(V^*\), required in ProcValidateCoreGP, are given by the functions \(\tilde{m}^*(\cdot)\) and \(\tilde{v}^*(\cdot,\cdot)\) which are given in ProcBuildEmulateDerivsGP. We can therefore validate an emulator of derivatives using the same procedure as that which we apply to validate an emulator of the core problem. It is often necessary, in response to the validation diagnostics, to rebuild the emulator using additional training runs which can of course, include derivatives. We hope to extend the validation process using derivatives as we gain more experience in validation diagnostics and emulating with derivative information.
The Bayes linear approach to emulating derivatives may be covered in a future release of the toolkit.
Tasks¶
Having obtained a working emulator, the MUCM methodology now enables efficient analysis of a number of tasks that regularly face users of simulators.
Prediction¶
The simplest of these tasks is to use the emulator as a fast surrogate for the adjoint, i.e. to predict what derivatives the adjoint would produce if run at a new point in the input space. The process of predicting function output at one or more new points for the core problem is set out in the prediction page ProcPredictGP. Here we are predicting derivatives and the process of prediction is the same as for the core problem. If the procedure in ProcBuildEmulateDerivsGP is followed, \(\tilde{D}, \tilde{t}, \tilde{A}, \tilde{e}\) etc. are used in replace of \(D, t, A, e\), as required in ProcPredictGP.
Sensitivity analysis¶
In sensitivity analysis the objective is to understand how the output responds to changes in individual inputs or groups of inputs. Local sensitivity analysis uses derivatives to study the effect on the output, when the inputs are perturbed by a small amount. Emulated derivatives could replace adjoint produced derivatives in this analysis if the adjoint is too expensive to execute or in fact does not exist.
Other tasks¶
Derivatives can be informative in optimization problems. If we want to find which sets of input values results in either a maximum or a minimum output then knowledge of the gradient of the function, with respect to the inputs, may result in a more efficient search. Derivative information is also useful in data assimilation.