(394i) Optimal Selection of Nonlinear Dynamic Sparse Surrogate Models with Exact Satisfaction of Mass, Energy and Thermodynamics Constraints

Data-driven model building of chemical process systems by using simulated/operational data has gained considerable interest as these models are often relatively easier to develop, maintain/update, and simulate real-time or even faster thus making them attractive for optimization, sensitivity and feasibility analysis, and online control. While artificial intelligence (AI) models are being significantly developed and utilized as data-driven models, models such as those developed by using artificial neural networks suffer from the lack of interpretability, possibility of unsatisfactory and unexpected predictions when the model is extrapolated, and large data requirement for training especially since these models are not necessarily sparse. To address the challenge of sparsity and interpretability, especially for dynamic sparse models, we developed the Bayesian Identification of Sparse Algebraic Model (BIDSAM)¹. Our approach also addressed limitations of many existing sparse identification approaches especially for dynamic systems by explicitly accounting for presence of noise in the data to avoid biased estimation of basis functions and model parameters when the training dataset is corrupted with noise.

Although the BIDSAM algorithm has been shown to attain superior performance compared to existing methods in terms of sparsity and predictive accuracy of models built from noisy data even in the presence of possible cross correlation, predictions from these models can violate conservation laws even when the training data satisfy the conservation laws. Although there are several approaches for incorporating physical laws into training of neural networks as a means of achieving mass/energy conservation^2–5, most of these are not generalizable and do not guarantee exact satisfaction of conservation laws. Also, many real processes are spatially distributed with possible temporal and/or spatial propagation of errors in measurement data due to time-varying uncertainties. In our previous work⁶, we developed an algorithm called MCBIDSAM, or mass-constrained BIDSAM for exact satisfaction of mass conservation by the BIDSAM models. This work extends our previous work by developing algorithms for BIDSAM that can exactly satisfy mass, energy and thermodynamics constraints simultaneously. Satisfying these constraints by data-driven models is very crucial for chemical processes for satisfactory extrapolative capabilities of the data-driven models and boundedness of predictions from such models.

We develop two algorithms. In the first algorithm, an optimization-based method is used for the outputs from the BIDSAM model to satisfy the conservation constraints for both steady-state and dynamic conditions. The BIDSAM algorithm includes an expectation maximization algorithm for Bayesian inference of the model parameter while a branch and bound algorithm is used for model building. This algorithm can be computationally expensive for large systems and can also lead to lower than desired predictive accuracy while still satisfying the conservation constraints. Therefore, we also develop another algorithm that exploits the structure and sparsity of the developed models to automatically construct and satisfy a set of equality constraints on the model parameters for the satisfaction of conservation laws.

The developed algorithms are applied to the modeling of the superheater section of an industrial boiler as well as a plug flow reactor (PFR) for acetone cracking coupled with a flash separator unit. For testing the algorithms, data are corrupted with noise of varying characteristics. Results show that the developed algorithms can learn dynamic sparse models with spatially and temporally distributed data with auto- and/or cross-correlated noise while exactly satisfying the mass, energy and thermodynamics constraints. Computational and predictive performances of both algorithms are also studied suggesting their suitability for various online and off-line applications.

References

1. Adeyemo, S. & Bhattacharyya, D. Optimal nonlinear dynamic sparse model selection and Bayesian parameter estimation for nonlinear systems. Comput. Chem. Eng. 180, 108502 (2024).
2. Karniadakis, G. E. et al. Physics-informed machine learning. Nat. Rev. Phys. 3, 422–440 (2021).
3. Sansana, J. et al. Recent trends on hybrid modeling for Industry 4.0. Comput. Chem. Eng. 151, 107365 (2021).
4. Bradley, W. et al. Perspectives on the integration between first-principles and data-driven modeling. Comput. Chem. Eng. 166, 107898 (2022).
5. Mukherjee, A. & Bhattacharyya, D. Development of Steady-State and Dynamic Mass and Energy Constrained Neural Networks for Distributed Chemical Systems Using Noisy Transient Data. Ind. Eng. Chem. Res. 63, 14211–14239 (2024).
6. Adeyemo, S. & Bhattacharyya, D. Sparse Mass-Constrained Nonlinear Dynamic Model Building from Noisy Data Using a Bayesian Approach. Ind. Eng. Chem. Res. (2025) doi:10.1021/acs.iecr.4c02481.