Certificates

We are aware of an issue with certificate availability and are working diligently with the vendor to resolve. The vendor has indicated that, while users are unable to directly access their certificates, results are still being stored. Certificates will be available once the issue is resolved. Thank you for your patience.

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

Authors: 
Olofsson, S., Imperial College
Misener, R., Imperial College
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.

References:

  • S. P. Asprey and S. Macchietto. Statistical tools for optimal dynamic model building. Comput Chem Eng, 24(2):1261–1267, 2000.
  • G. E. P. Box and W. J. Hill. Discrimination among mechanistic models. Technometrics, 9(1):57–71, 1967.
  • G. Buzzi-Ferraris, P. Forzatti, and P. Canu. An improved version of a sequential design criterion for discriminating among rival multiresponse models. Chem Eng Sci, 45(2):477–481, 1990.
  • B. H. Chen and S. P. Asprey. On the design of optimally informative dynamic experiments for model discrimination in multiresponse nonlinear situations. Ind Eng Chem Res, 42(7):1379–1390, 2003.
  • D. Espie and S. Macchietto. The optimal design of dynamic experiments. AIChE J, 35(2):223–229, 1989.
  • W. G. Hunter and A. M. Reiner. Designs for discriminating between two rival models. Technometrics, 7(3):307–323, 1965.
  • A. Mesbah, S. Streif, R. Findeisen, and R. D. Braatz. Active fault detection for nonlinear systems with probabilistic uncertainties. In Proceedings of the 19th World Congress of the International Federation of Automatic Control (IFAC), pages 7079–7084, Cape Town, South Africa, 2014.
  • C. Michalik, M. Stuckert, and W. Marquardt. Optimal experimental design for discriminating numerous model candidates: The AWDC criterion. Ind Eng Chem Res, 49:913–919, 2010.
  • S. Olofsson, L. Hebing, S. Niedenführ, M. P. Deisenroth, and R. Misener. GPdoemd: A Python package for design of experiments for model discrimination. Comp Chem Eng, 125:54–70, 2019.
  • E. G. Ryan, C. C. Drovandi, and A. N. Pettitt. Fully Bayesian experimental design for pharmacokinetic studies. Entropy, 17:1063–1089, 2015.
  • E. G. Ryan, C. C. Drovandi, J. M. McGree, and A. N. Pettitt. A review of modern computational algorithms for Bayesian optimal design. Int Stat Rev, 84:128–154, 2016.
  • S. Streif, F. Petzke, A. Mesbah, R. Findeisen, and R. D. Braatz. Optimal experimental design for probabilistic model discrimination using polynomial chaos. In Proceedings of the 19th World Congress of the International Federation of Automatic Control (IFAC), pages 4103–4109, Cape Town, South Africa, 2014.
  • J. Vanlier, C. A. Tiemann, P. A. J. Hilbers, and N. A. W. van Riel. Optimal experiment design for model selection in biochemical networks. BMC Syst Biol, 8(20), 2014.