Modelling Chemical Reaction Networks using Neural Ordinary Differential Equations

TL;DR

This work tackles the challenge that ODEs derived from chemical reaction networks (CRNs) via mass-action can be incomplete due to hidden dynamics. It introduces neural ODEs (nODEs) that augment the theoretical CRN vector field with a data-driven term, yielding a hybrid model $\dfrac{dy}{dt}=h_\kappa(t,y)+f_\theta(t,y)$ that better fits experimental concentration trajectories and reveals missing interactions. Through experiments on an oscillatory CRN, the authors show that nODEs improve predictive accuracy for both aperiodic and oscillatory data and enable interpretation of residual dynamics, including the transfer of learned dynamics across experiments. While the nODE often surpasses the purely theoretical model in period estimation and residual correction, it may not reliably classify oscillation regimes without more data, underscoring the value of jointly guiding experiments and model refinement. The approach provides a practical framework to diagnose model misspecification, quantify missing reactions, and inform the design of future CRNs with improved dynamic fidelity.

Abstract

In chemical reaction network theory, ordinary differential equations are used to model the temporal change of chemical species concentration. As the functional form of these ordinary differential equations systems is derived from an empirical model of the reaction network, it may be incomplete. Our approach aims to elucidate these hidden insights in the reaction network by combining dynamic modelling with deep learning in the form of neural ordinary differential equations. Our contributions not only help to identify the shortcomings of existing empirical models but also assist the design of future reaction networks.

Figures (11)

• Figure 1: The predictive performance of the nODE on the single-pulse data. (a), the experimental measurements (solid blue), predictions from the nODE (dashed red) and $h_\kappa$ (dashdotted green). The four panes illustrate the molar concentrations for the species dibenzofulvene, Fmoc-piperidine, piperidine and N-acetyl piperidine. (b), the neural network contribution for the four measured species.
• Figure 2: The predictive performance of the nODE for which the reaction for N-acetyl piperidine is hidden. (a), the experimental measurements (solid blue), predictions from the nODE (dashed red) and $h_\kappa$ (dashdotted green) for N-acetyl piperidine. (b), the neural network contribution.
• Figure 3: The predictive performance of the nODE on the oscillating open system for dibenzofulvene. (a), the experimental measurements (solid blue), predictions from the nODE (dashed red) and $h_\kappa$ (dashdotted green). (b), the neural network contribution.
• Figure 4: The oscillation spaces predicted by (a) the theoretical model, (b) the nODE trained on the complete time series and (c) the nODE trained on the first half of the time series. The coloured regions show the predicted period of the oscillations. Outside of these regions, the models predict damped or no oscillations. The filled squares indicate experimentally observed sustained oscillations, the open squares are associated with damped oscillations catalyticOscillator.
• Figure 5: The difference between the predicted oscillation spaces. (a), the predictions of the nODE trained on the full time series minus the theoretical predictions. (b), the predictions of the nODE trained on the full time series minus those of the nODE trained on half of the time series. (c) the predictions of the nODE trained on half of the time series minus the theoretical predictions.