(369e) Optimal Bayesian Experiment Design for Constrained Nonlinear Dynamic Systems Using Stochastic Surrogate Models

The optimal design of experiments is crucial for maximizing the information content of observations across a wide-range of experimental goals. Model-based optimal experiment design (OED) uses a mathematical relationship between the design variables, parameters, and observables of a system to systematically select experimental conditions that maximize some design metric that is relevant to, for example, parameter estimation or model discrimination [1]. Classical OED design criteria are generally defined in terms of some scalar metric of the Fisher information matrix (FIM) such as the alphabetic optimality criteria A-, D-, and E-optimality [2]. Alternatively, OED can adopt a Bayesian perspective, where the design criteria are expressed in terms of an expected utility quantity that accounts for both prior and posterior uncertainty in the model parameters from a decision theoretic point-of-view [3]. Although the expected utility reduces to mathematical forms that are equivalent to their classical FIM counterparts in the case of linear models subject to Gaussian uncertainty, analytic expressions for the expected utility are not available for nonlinear models. Tractable design criteria can be derived for nonlinear models by introducing additional assumptions that often include linearization of the model, Gaussian approximations of the posterior distribution, as well as restrictions on the marginal data distribution. Some of these approximations result in prior expectations over the FIM, though cruder nominal OED methods that maximize a function of the FIM evaluated at a current best guess for the parameters are commonly applied for computationally-demanding models [4]. However, these methods are not suitable when the parameter distribution is broad or deviates significantly from normality whereas the Bayesian design framework is completely free from these limiting assumptions [5].

This work presents a Bayesian OED approach for nonlinear dynamic systems (possibly involving high-dimensional design spaces) subject to state chance constraints [6]. Due to the complicated form of the expected utility, it must be estimated using sample-based methods and, in particular, a nested Monte Carlo (MC) estimator that is very expensive to evaluate for the full dynamic model. Inspired by [7], we propose the use of a surrogate model to overcome the challenge of repeatedly evaluating the expected utility and its gradients within numerical optimization. The proposed surrogate modeling approach is based on recently developed arbitrary polynomial chaos (aPC) theory [8,9]. Not only does aPC apply to arbitrary parameter distributions (e.g., correlated or multi-modal distributions), but we show how the expansion can be locally constructed around each design visited during the optimization procedure from a minimal set of model evaluations, which enables a simple way to tradeoff accuracy and computational cost. We also demonstrate how chance constraints can be incorporated into the optimization problem using the aPC-based surrogate model. A key feature of our Bayesian OED method is that it can be directly implemented with state-of-the-art dynamic optimization methods (e.g., multiple shooting or collocation [10]) so that the underlying structure of the optimization can be exploited for efficiency in a similar manner to classical OED. We demonstrate the effectiveness of the proposed method on a benchmark predator-prey problem. Simulation results show that we can significantly lower the computational cost of Bayesian OED compared to the approach in [7], while simultaneously improving the solution accuracy. In addition, we demonstrate that the cost due to the proposed chance constraint handling method is negligible compared to the main cost of estimating the expected utility.

References

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[6] J. A. Paulson, M. Martin-Casas, and A. Mesbah, “Optimal Bayesian experiment design for nonlinear dynamic systems with chance constraints,” Journal of Process Control, 2019 (in press).

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