FMEnets: Flow, Material, and Energy networks for non-ideal plug flow reactor design

TL;DR

FMEnets introduces a physics-informed machine learning framework that couples Navier-Stokes flow, material balances, and energy balance to model non-ideal plug flow reactors. It features three interconnected sub-networks and supports forward and inverse problems via two representation modes: FME-PINNs (MLPs) and FME-KANs, with sequential training and multiple enhancements to improve stability. Across three reaction-case scenarios, FMEnets achieves relative errors below 2.5% for unknown kinetic parameters and demonstrates robustness to noisy data, especially with FME-KANs. The approach offers a computationally efficient, flexible design tool capable of integrating sparse experimental data and empirical correlations to guide reactor design and optimization.

Abstract

We propose FMEnets, a physics-informed machine learning framework for the design and analysis of non-ideal plug flow reactors. FMEnets integrates the fundamental governing equations (Navier-Stokes for fluid flow, material balance for reactive species transport, and energy balance for temperature distribution) into a unified multi-scale network model. The framework is composed of three interconnected sub-networks with independent optimizers that enable both forward and inverse problem-solving. In the forward mode, FMEnets predicts velocity, pressure, species concentrations, and temperature profiles using only inlet and outlet information. In the inverse mode, FMEnets utilizes sparse multi-residence-time measurements to simultaneously infer unknown kinetic parameters and states. FMEnets can be implemented either as FME-PINNs, which employ conventional multilayer perceptrons, or as FME-KANs, based on Kolmogorov-Arnold Networks. Comprehensive ablation studies highlight the critical role of the FMEnets architecture in achieving accurate predictions. Specifically, FME-KANs are more robust to noise than FME-PINNs, although both representations are comparable in accuracy and speed in noise-free conditions. The proposed framework is applied to three different sets of reaction scenarios and is compared with finite element simulations. FMEnets effectively captures the complex interactions, achieving relative errors less than 2.5% for the unknown kinetic parameters. The new network framework not only provides a computationally efficient alternative for reactor design and optimization, but also opens new avenues for integrating empirical correlations, limited and noisy experimental data, and fundamental physical equations to guide reactor design.

Figures (8)

• Figure 1: (a) Schematic of the tubular reactor with external coolant; (b) Modeled domain and boundary conditions; and (c) Sampling/measurement points at different residence times along the reactor.
• Figure 2: Schematic of the FMEnets architecture for solving the forward problem. Three neural networks ($NN_1, NN_2, NN_3$) each enforce different governing equations, including Navier–Stokes (NS), material balance (MB), and energy balance (EB), along with boundary conditions and inlet/outlet experimental data. $NN_1$ first predicts the velocity field ($u, v$), which then feeds into $NN_2$ and $NN_3$ to learn species concentrations and temperature, respectively. Each sub-network ($NN_1, NN_2, NN_3$) can be implemented as either a KAN or an MLP.
• Figure 3: Schematic of the FMEnets architecture for the inverse problem with a two‐step procedure. Step 1: We train $NN_1$ to enforce the Navier–Stokes (NS) equations and predict the velocity field ($u,v$). Step 2: $NN_2$ and $NN_3$ employ those velocity predictions to solve the material (MB) and energy balance (EB) equations with boundary conditions and multi‐residence‐time experimental data while inferring the unknown activation energy $E_a$. Each sub-network ($NN_1, NN_2, NN_3$) can be implemented as either a KAN or an MLP.
• Figure 4: Distribution of 6-point Gauss quadrature points on a reference triangle and exact integration for polynomial degree of 4.
• Figure 5: Visualization of the accuracy of the FME-PINNs against Finite Element Method (FEM) simulations for Case 1: FEM (left column), FME-PINNs (middle column), and the absolute error (right column) for each variable, including velocity components ($u, v$) in $m/s$, pressure ($p$) in $Pa$, species concentrations ($C_a, C_b$) in $mol/m^3$, and temperature ($T$) in $K$.