Physics-informed neural networks need a physicist to be accurate: the case of mass and heat transport in Fischer-Tropsch catalyst particles

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as an influential technology, merging the swift and automated capabilities of machine learning with the precision and dependability of simulations grounded in theoretical physics. PINNs are often employed to solve algebraic or differential equations to replace some or even all steps of multi-stage computational workflows, leading to their significant speed-up. However, wide adoption of PINNs is still hindered by reliability issues, particularly at extreme ends of the input parameter ranges. In this study, we demonstrate this in the context of a system of coupled non-linear differential reaction-diffusion and heat transfer equations related to Fischer-Tropsch synthesis, which are solved by a finite-difference method with a PINN used in evaluating their source terms. It is shown that the testing strategies traditionally used to assess the accuracy of neural networks as function approximators can overlook the peculiarities which ultimately cause instabilities of the finite-difference solver. We propose a domain knowledge-based modifications to the PINN architecture ensuring its correct asymptotic behavior. When combined with an improved numerical scheme employed as an initial guess generator, the proposed modifications are shown to recover the overall stability of the simulations, while preserving the speed-up brought by PINN as the workflow component. We discuss the possible applications of the proposed hybrid transport equation solver in context of chemical reactors simulations.

Figures (10)

• Figure 1: Schematic representation of the catalytic particle as an element of chemical reactor performing Fischer-Tropsch synthesis. The mass and heat transport in the particle is modeled by diffusion and heat transfer equations, in which the source terms result from the kinetics of underlying reactions.
• Figure 2: $H_2$ consumption rate $- s (c)$ as a function of $H_2$ concentration $c$ (at $P_{CO} = 1 \; \mathrm{MPa}$, $P_{H_2 O} = 0.5 \; \mathrm{MPa}$, $T = 493.15 \; \mathrm{K}$). The difference between exact and approximated dependencies might seem negligible when characterized by the absolute difference (and also hardly noticeable in linear scale, a), but exhibits drastically different asymptotic behaviour (best viewed in logarithmic scale, b).
• Figure 3: Exact dependence of the source term $s$ on substance concentration $c$ and two approximations which differ from it only in a small region (zoomed in a right plane) near $c = 0$
• Figure 4: The concentration profiles $c (x)$ obtained with exact form of dependence of the source term $s$ on concentration $c$ and two its approximations (\ref{['s1approx-toy']}) and (\ref{['s2-approx-toy']}). Right plane explicitly shows these approximations for the range of concentration values requested by the numerical solver. For convenience, threshold values of $c = 0$ and $c = \theta$ are shown in a left plane with blue and red the dotted lines respectively.
• Figure 5: Dependence of $[S]$ on $P_{H_2}$ at $T = 493.15 K$ under different values of $P_{\operatorname{CO}}$ and $P_{H_2 O}$ obtained using numerical equation solver for \ref{['S-equation-c0-cS']} with microkinetics parameters taken from Todic-CO-insertion-2014Todic-corrigendum-2015. Right panel additionally shows the ratio $\frac{[S]}{(P_{H_2})^2}$ for convenience.

Theorems & Definitions (2)

• proposition 1
• proof