(579c) A Novel Hybrid Modeling Framework: Dynamic Estimation of Spatiotemporal Parameters in Complex Chemical Processes Governed By PDEs

In chemical engineering and applied sciences, accurately modeling complex systems governed by partial differential equations (PDEs) remains a significant challenge. These systems, with their spatiotemporally varying parameters and dynamic boundaries, require sophisticated approaches beyond traditional models, which often fail to capture their latent dynamics. The exploration of hybrid modeling, which combines machine learning (ML) with classical mathematical models, has shown promise in overcoming these limitations [1,2]. Hybrid models can adaptively refine estimates based on observed data, improving fidelity to real-world phenomena. However, the accurate estimation of spatiotemporally varying parameters in complex PDE systems has been an unresolved challenge, with traditional methods entailing a heavy computational load [3]. Addressing this gap, our work introduces a novel hybrid modeling framework employing convolutional neural networks (CNNs) for dynamic parameter estimation in a reaction-diffusion system, demonstrating a significant stride towards the challenge of accurately capturing latent chemical mechanisms through estimation of spatiotemporally varying parameters [4].

To this end, we present an approach to extend the current hybrid modeling methodology to complex systems governed by PDEs where latent spatiotemporal parameters are present. This synergy allows for the dynamic estimation and update of spatiotemporal model parameters, a significant advancement over static or temporal parameter models. Model training is executed through a back-propagation algorithm, adept at efficiently updating these parameters while satisfying the Courant−Friedrichs−Lewy (CFL) numerical stability criterion. This work focuses on reaction-diffusion processes, which are often employed to model phenomena such as tumor growth, where the dissemination of cancer cells in tissues and their proliferation can be elucidated using reaction−diffusion equations [5,6]. Convolutional neural networks (CNNs) are utilized to uncover relationships between spatiotemporally varying inputs, such as cell density, and parameters that vary both spatiotemporally, like diffusivity, and temporally like cell proliferation rates and carrying capacity density which remain obscured and difficult to estimate directly from experimental observations.

Through rigorous model training and validation against experimental data generated through running the Porous Fisher model, the hybrid model is proven to significantly outperform traditional first-principles models, particularly in its ability to predict diffusivity across space and time with remarkable precision. This work also addresses the challenges and solutions in maintaining numerical stability during model training. This includes strategies for maintaining a lower learning rate and introducing a novel concept of window size for inputs to ensure stable and accurate model predictions. The validation of the hybrid model's efficacy is underscored by a comparative analysis of its performance against conventional models, where the hybrid model has a mean squared error (MSE) of dramatically lower than that of first-principles model with an MSE of . This outcome not only validates the model's robustness but also its potential as a powerful tool for probing into the latent mechanisms governing complex systems. In conclusion, this proposed hybrid modeling approach offers a novel solution to a longstanding challenge in modeling complex systems, providing deeper insights into the spatiotemporal dynamics of reaction-diffusion processes and beyond. This work not only enhances our understanding of latent chemical mechanisms but also opens new horizons for the development of more accurate and reliable hybrid models across many disciplines where PDEs are foundational.

References:

1. Bangi, Mohammed Saad Faizan, and Joseph Sang-Il Kwon. "Deep hybrid modeling of chemical process: Application to hydraulic fracturing." Computers & Chemical Engineering 134 (2020): 106696.
2. Shah, Parth, et al. "Deep neural network-based hybrid modeling and experimental validation for an industry-scale fermentation process: Identification of time-varying dependencies among parameters." Chemical Engineering Journal 441 (2022): 135643.
3. Xun, Xiaolei, et al. "Parameter estimation of partial differential equation models." Journal of the American Statistical Association 108.503 (2013): 1009-1020.
4. Pahari, Silabrata, Parth Shah, and Joseph Sang-Il Kwon. "Unveiling Latent Chemical Mechanisms: Hybrid Modeling for Estimating Spatiotemporally Varying Parameters in Moving Boundary Problems." Industrial & Engineering Chemistry Research 63.3 (2024): 1501-1514.
5. Warne, David J., Ruth E. Baker, and Matthew J. Simpson. "Using experimental data and information criteria to guide model selection for reaction–diffusion problems in mathematical biology." Bulletin of Mathematical Biology 81.6 (2019): 1760-1804.
6. Mei, Ming, and Yong Wang. "Remark on stability of traveling waves for nonlocal Fisher-KPP equations." Int. J. Numer. Anal. Model. Ser. B 2.4 (2011): 379-401.