Evaluation of optimisation and Bayesian inference methods for reaction rates in atmospheric chemical mechanisms
Authors: Valery Ashu, Wenqing Peng, Zhi-Song Liu, Heikki Haario, Andreas Rupp, Taiwo Ashu, Petri Clusius, Lukas Pichelstorfer, +2 more
Organizations: Department of Computational Engineering, LUT University, 15210, Lahti, Finland · Atmospheric Modelling Center-Lahti, 15140, 15140, Lahti, Finland · Institute for Atmospheric and Earth System Research / Physics, Faculty of Science, University of Helsinki, 00014, Helsinki, Finland · Saarland University, Campus E1.1, 66123, Saarbrücken, Germany · pi-numerics, Neumarkt am W., 5202, Austria
Constraining reaction rate coefficients is a central challenge in the development of explicit atmospheric chemical mechanisms, particularly for autoxidation systems where many reaction pathways are only indirectly observed through high-resolution mass spectrometry. In this study, we evaluate rate-coefficient optimisation methods for a toy-case autoxidation mechanism using synthetic data with known ground truth. Two complementary approaches are compared: ODE-constrained neural-network optimisation, which provides efficient point estimates of uncertain rate coefficients, and the Markov Chain Monte Carlo (MCMC) approach, which samples the posterior distribution of rate coefficients and quantifies parameter uncertainty. The methods are tested using direct concentration observations and mass-spectral observations under different noise levels. For unperturbed and low-noise synthetic observations, both methods converged towards the known rate coefficients, with the neural-network optimiser providing faster point estimates. Under high-noise conditions (with the signal-to-noise ratio approximately S / N = 1), however, MCMC was substantially more robust in recovering the rate coefficients. The posterior analysis shows that mass-spectral aggregation broadens credible intervals even at low noise, and that high-noise mass spectra can leave many individual reaction rates weakly identifiable. Posterior predictive validation nevertheless shows how broad parameter uncertainty constrained by MCMC remains consistent with accurate reproduction of the observable mass spectrum. These results demonstrate that point-estimation and Bayesian sampling methods provide complementary information: neural-network optimisation is effective for informative data, whereas MCMC is essential for diagnosing uncertainty, non-uniqueness, and identifiability in noisy or aggregated inverse problems.
Figures & tables
Measured data
Baseline
noise-free case
low-noise case
high-noise case
MCMC
NN
MCMC
NN
MCMC
NN
Concentration
0.0928
0.0120
0.000147
0.0146
0.0114
0.0768
0.884
Mass spectrum
0.0928
0.0780
0.0402
0.0930
0.0746
0.2921
1.306
Table 1: Overall RMSRE comparison of MCMC and Neural Network across noise levels. The MCMC values are computed from posterior median rate estimates. The baseline RMSRE denotes the value between the initial and true coefficient estimates.
Figure 1: Log relative error of estimated rate coefficients
Figure 2: Optimisation convergence. Panels 2(a) and 2(c) show the trajectory-fitting objective, and panels 2(b) and 2(d) the corresponding error of the estimated rate coefficients. Solid lines use a single noisy realisation; dashed lines use stochastic optimisation over n=4 noisy samples.
Baseline
Low-noise
High-noise
n=1
n=4
n=1
n=4
Conc.
0.0928
0.0114
0.00367
0.884
N/A
Mass
0.0928
0.0746
0.0478
1.306
0.289
Table 2: RMSRE of estimated rate coefficients with additional noise samples.
Figure 3: Comparison of optimisation against ODE trajectory and finite difference derivative, using 0% noise concentration observations. t is the number of time steps sampled from the trajectories.
Case
95% CI width
Median range
Coverage
Concentration
0.015
0.937–1.069
45/50
Mass spectra, low noise
0.047
0.703–1.246
47/50
Mass spectra, high noise
0.613
0.294–2.472
42/50
Table 3: Posterior constraints from the three DRAM-MCMC cases after discarding the first 20 % of each chain as burn-in. The CI width is reported as the median parameter-wise 95 % credible-interval width in log10 multiplier space. The posterior median range is reported in rate-multiplier space. Coverage gives the number of inferred parameters whose 95 % credible interval contains the true multiplier value of one.
Figure 4: Sorted widths of the parameter-wise 95 % credible intervals for the three MCMC cases. Direct concentration observations yield the narrowest posterior intervals. Mass-spectral aggregation broadens the posterior under low-noise conditions, while the high-noise mass-spectral case produces substantially weaker constraints for many parameters.
Figure 5: Posterior medians and 95 % credible intervals for all 50 rate multipliers. The dashed line denotes the true multiplier value of one. The concentration and low-noise mass-spectral cases constrain most parameters tightly around the true value, whereas the high-noise mass-spectral case produces broad intervals and stronger parameter-to-parameter variation in posterior uncertainty.
Figure 6: Representative marginal posterior distributions for selected rate multipliers in log10 space. The vertical dashed line marks the true value. The examples illustrate the transition from sharply localized posteriors in the concentration and low-noise mass-spectral cases to broader and more weakly identified posteriors in the high-noise mass-spectral case.
Figure 7: Posterior predictive check for the high-noise mass-spectral case. The black crosses show the noisy observation, the blue open circles show the clean synthetic spectrum, and the brown markers show the posterior mean prediction from 200 posterior samples. Error bars denote the 90 % posterior predictive interval for each nominal mass channel.
Department of Statistical Sciences, University of Toronto, Toronto, Ontario, Canada · Schools of Mathematics & Physics and Chemistry & Molecular Biosciences, The University of Queensland, St Lucia, Queensland, Australia · Australian Research Council Centre of Excellence in Quantum Biotechnology, St Lucia, Queensland, Australia +9
State Key Laboratory of Fine Chemicals, Frontier Science Center for Smart Materials, Dalian University of Technology, Dalian, 116024, China · Dalian Key Laboratory of Intelligent Chemistry, CR Belt and Road Joint Laboratory on Intelligent Chemistry and Advanced Materials of Liaoning Province, School of Chemistry, Dalian University of Technology, Dalian, 116024, China
Shanghai Jiao Tong University, Shanghai, China · Shanghai Innovation Institute, Shanghai, China · Shanghai Artificial Intelligence Laboratory, Shanghai, China