Neural networks are widely used as surrogate models but they do not guarantee physically consistent predictions thereby preventing adoption in various applications. We propose a method that can enforce NNs to satisfy physical laws that are nonlinear in nature such as enthalpy balances. Our approach, inspired by Picard successive approximations method, aims to enforce multiplicatively separable constraints by sequentially freezing and projecting a set of the participating variables. We demonstrate our PicardKKThPINN for surrogate modeling of a catalytic packed bed reactor for methanol synthesis. Our results show that the method efficiently enforces nonlinear enthalpy and linear atomic balances at machine-level precision. Additionally, we show that enforcing conservation laws can improve accuracy in data-scarce conditions compared to vanilla multilayer perceptron.

Multiscale modeling of catalytical chemical reactors typically results in solving a system of partial differential equations (PDEs) or ordinary differential equations (ODEs). Despite significant progress, the numerical solution of such PDE or ODE systems is still a computational bottleneck. In the past, deep learning techniques have gained attention for developing surrogate models in chemical engineering. Also, hybrid models and physics-informed neural networks (PINNs) have been developed to integrate physical knowledge and data-driven approaches. However, it is often unclear how such modeling approaches compare for specific case studies. In this study, we investigate and compare state-of-the-art surrogate and hybrid models for the spatial evolution of the state variables in a packetbed reactor for methanol production. Firstly, we develop a tailored hybrid model based on PINNs, thereby seamlessly integrating physical knowledge and data. Secondly, we investigate a recently-developed time-series transformer model to learn the spatial evolution of the state variables. As a benchmark model, we train a traditional multilayer perceptron (MLP) and compare the models to a standard numerical integration technique. We achieve orders of magnitude in speedup using MLPs and PINNs when compared to classical ODE solvers, while maintaining high levels of accuracy in modeling the underlying system.