(324f) Grey-Box Identification for a Class of Nonlinear Dynamical Systems

We build upon early efforts towards “physics-informed” neural network modeling [1,2] and expand the class of systems for which these methods apply. Artificial neural networks (ANNs) are powerful models for universal function approximation, which has led to their use in a growing number of domains, including image processing, dynamic modeling, time series analysis, and more. Within the field of process systems engineering, ANNs are particularly useful because they can be used for modeling important nonlinear phenomena, such as bifurcations or saturations, which cannot be represented using linear systems. One disadvantage, however, for ANNs (especially for modern “deep” ANNs) is that they require a large quantity of data on which to train the model. This data is often unavailable in the domains of scientific research and industrial systems, where system identification experiments are costly and time-consuming. Moreover, ANNs are black box models that provide no structure and utilize no physical insight about the system. The use of such a black box model that is not well characterized for all possible operating conditions can result unexpected, unrealistic, or possibly dangerous model outputs (if, for example, the model is used for predictive control of a hazardous process). To alleviate these concerns, we present a class of grey-box artificial neural network models (GB-ANNs).

In this work, we introduce a new framework for designing GB-ANNs for systems constrained by conservation laws governed by unknown constitutive relationships and unknown state variables. We use delay embeddings to reconstruct unknown system states (c.f., the Takens embedding theorem [3,4]) and modern ANN architectures to represent the constitutive laws. After training the GB-ANNs on (a modest amount of) representative data, the resulting model is a delay differential equation that can be used for state estimation, model predictive control, or other on-line monitoring tasks. We demonstrate our methodology using three numerical examples: (1) a CSTR with an enzymatic reaction, (2) a continuous bioreactor, and (3) a CSTR with a catalytic surface reaction, and we compare the performance of the GB-ANNs with black box models for tasks such as off-line prediction and on-line state estimation.

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[3] Takens, F. Detecting strange attractors in turbulence. In Dynamical systems and turbulence, Warwick 1980, pp. 366-381. Springer, Berlin, Heidelberg, 1981.

[4] Stark, J. Delay Embeddings for Forced Systems. I. Deterministic Forcing. J. Nonlinear Sci. 9, 255–332 (1999).