Integrating Deep Neural Networks for Hybrid Modeling of Complex Chemical Processes: Estimation of Spatiotemporally Varying Parameters in Moving Boundary Problems

Hybrid modeling has gained substantial traction due to its capacity to combine machine learning techniques with the preservation of the model's physical essence. While these hybrid models have primarily tackled temporal processes governed by ordinary differential equations (ODEs), the complexity of many real-world systems - mirroring diverse physical processes - surpasses this scope. This work examines hybrid modeling methods extensively applied to an intricate biological system governed by the reaction-diffusion equation, which is a partial differential equation (PDE), targeting the challenges arising from latent chemical mechanisms. The methodology introduces a hybrid modeling architecture that synergizes neural networks and mathematical methods to estimate varying parameters across both space and time within a class of moving boundary problems like reaction-diffusion, all while upholding boundary conditions. The training of the model is done using a backpropagation algorithm that efficiently updates these parameters while ensuring numerical stability. The hybrid model is applied to Reaction-Diffusion models, and the results discuss the accurate estimation of spatiotemporally varying diffusivity, and temporally varying cell proliferation rate and cell carrying density.