(118h) Design of Dynamic Experiments for Model Discrimination Under Uncertainty Using Gaussian Process Surrogate Models

Chemical engineering often deals with noisy, uncertain, and dynamic processes. Modelling these processes is difficult, and exacerbated by the difficulty of observing mechanisms and reactions on the molecular level. Researchers and engineers can hypothesise several mechanistic models for an underlying system mechanism, but may have insufficient experimental data to discriminate between the hypotheses, i.e. discard inaccurate models. Given a set of rival models and initial experimental data, optimal experimental design suggests conditions maximising the expected utility from additional experiments to aid model discrimination [Espie and Macchietto, 1989; Asprey and Macchietto, 2000]. Because the model parameters are tuned to fit the models to noisy experimental data, the optimisation procedure needs to account for both parametric uncertainty and measurement noise [Box and Hill, 1967; Chen and Asprey, 2003].

Researchers have considered optimal design of experiments for model discrimination for over 50 years [Hunter and Reiner, 1965]. Most existing research can be divided into two approaches: (i) the classical analytic approach and (ii) the data-driven approach. The analytic approach, e.g. Box and Hill [1967], Buzzi-Ferraris et al. [1990] and Michalik et al. [2010], use first-order Taylor approximations to propagate Gaussian uncertainty from input to output. The next experiment is chosen by maximising closed-form expressions for the divergence between the rival predictive distributions. The analytic approach is computationally efficient. But the analytic approach depends on derivative information which may not always be readily available. The alternative, data-driven approach, e.g. Vanlier et al. [2014] and Ryan et al. [2015], samples from the input distribution to generate output samples through simulation. The next experiment is chosen by approximating the divergence between the resulting samples. The data-driven approach is flexible with regards to the models, effectively treating them as black boxes. However, the computational cost of approximating the distributions' divergence across the design space is often prohibitive [Ryan et al., 2016].

Olofsson et al. [2019] present a third alternative to the analytic and data-driven approaches: the hybrid approach, where the original models are treated as black boxes and used to train Gaussian process surrogate models that can be used in an analytic fashion to find the next experiment. The hybrid approach combines flexibility with regards to the original models, with the computational tractability of the analytic approach. The surrogate models map the full input space directly to the final observations. Hence, the Olofsson et al. [2019] approach works well for models with low-dimensional input spaces.

Espie and Macchietto [1989] extend the analytic approach to discriminate between dynamic (time-dependent) models. To the best of our knowledge, there is no existing work discriminating dynamic black-box models. In dynamic models, the control sequence yields a high-dimensional input space. Hence, the data-driven approach and Olofsson et al. [2019] hybrid approach may be too computationally demanding because they suffer from the curse of dimensionality associated with developing accurate surrogate approximations of the rival mechanistic models.

This presentation assumes that the system dynamics are described by rival continuous- or discrete-time state-space models, with black-box latent state transition functions. Experimental observations consist of a subset of the latent states. We propose replacing the original transition functions with Gaussian process surrogates, which can be used in an analytic fashion. Using the surrogates, we marginalise out uncertainty in variables and parameters with approximate inference, to produce Gaussian predictive distributions.

In static models, the measurement noise and parametric uncertainty are often the only sources of uncertainty treated formally [Chen and Asprey, 2003]. For dynamic (time-dependent) models, Streif et al. [2014] and Mesbah et al. [2014] have also considered uncertainty in the latent states. Our work extends the classical and hybrid approaches for optimal design of dynamic experiments to account for uncertainty in control inputs and uncertainty in the latent states due to process noise.

Results show these contributions working well for both classical and new test instances, with both continuous- or discrete-time black-box state space models.

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