Evaluation of the degree of rate control via automatic differentiation
Yilin Yang, Siddarth K. Achar and John R. Kitchin, “Evaluation of the degree of rate control via automatic differentiation”, AIChE Journal 68(6), e17653 (2022). https://doi.org/10.1002/aic.17653
This summary was written by a language model from the paper and has not been checked line by line. Rely on the published paper for details.
Question
Campbell’s degree of rate control (DRC) measures how strongly each elementary step of a reaction network controls a target rate: it is the derivative of the log of the net rate with respect to the log of a forward rate constant, holding the step’s equilibrium constant and all other rate constants fixed (a thermodynamic variant differentiates with respect to intermediate free energies). In practice the derivatives are usually taken by finite differences (FD), which need a careful choice of perturbation size to balance truncation error against floating point rounding error, and which cost one extra simulation per parameter; the authors cite a CO2 hydrogenation sensitivity study that needed about 300 CPU-hours. Direct and adjoint sensitivity methods give derivatives of the ODE solution but still leave the derivatives of the rate expression to be assembled. The paper asks whether automatic differentiation (AD) can give accurate DRCs, steady state and transient, more cheaply and with less effort (Yang et al. 2022).
Approach
The paper first explains forward and reverse mode AD on a small two-variable function, showing the computational graph and the derivative propagation in each mode, and notes that forward mode suits few inputs and reverse mode few outputs. It then shows how AD propagates through an ODE integrator (illustrated with forward Euler), so that derivatives of a time-dependent solution and of any post-processing such as a rate expression are obtained end to end. Citing published benchmarks, the authors note forward AD has the same O(np) scaling as FD but is faster for small ODE systems (under about 100 parameters) because derivatives are computed alongside function values. The implementation uses Julia’s ForwardDiff with DifferentialEquations, which provides differentiable stiff and non-stiff solvers (Yang et al. 2022).
The transient DRC used here perturbs a parameter at time zero and asks how the response at time t changes; Campbell’s alternative transient definition, which perturbs at time t, is treated in the supporting information. For uncertain parameters they adopt the distributed evaluation of local sensitivity analysis (DELSA), a hybrid local-global method that evaluates local sensitivities at many samples of the parameter space and normalizes them by a first-order estimate of output variance, giving a distribution of importance for each parameter at far lower cost than Sobol indices (Yang et al. 2022).
Three microkinetic cases were studied: (I) Foley and Bhan’s hypothetical two-step reaction, which has an analytical transient solution; (II) a seven-step redox water-gas shift mechanism from Motagamwala and Dumesic; and (III) a 17-step propylene partial oxidation mechanism involving surface oxygen vacancies, from a DFT and microkinetic study by Song and Wang.
Key findings
In case I the AD transient DRCs matched the analytical expressions over the whole time range: step 1 starts at 0 and rises to 1, step 2 starts at 1 and decays to 0, and the sum of DRCs equals one only at steady state. In case II, at the stated gas pressures, AD reproduced the original result that only two steps matter: OH* dissociation (DRC 0.88) and CO2* formation (DRC 0.12) (Yang et al. 2022).
In case III (500 K, 0.1 bar propylene, 0.05 bar O2, rate of propylene oxide desorption), propylene adsorption in the type III configuration and PO2* desorption had the largest positive DRCs, while O2* desorption was inhibiting. At 350 K, FD with perturbations of 1e-4, 1e-11 and 1e-14 was compared with AD for steps 3, 4, 14 and 17. The best FD step (1e-11) agreed with AD to within about 1%, the large step had errors up to 7%, and the tiny step was dominated by rounding error (845% average deviation for step 4). The transient DRC took about 1.0 s with FD and 0.1 s with AD (Yang et al. 2022).
For DELSA, the forward rate constants in case III were given an uncertainty equivalent to plus or minus 2.9 kJ/mol in activation energy, with 200 quasi-random Sobol samples. Only steps 3, 14 and 17 had average importance above 0.1, the same set flagged by the DRC, but the distributions added information: step 14 was bimodal, with roughly half the samples near 0 and half near 1 (Yang et al. 2022).
Significance
The paper makes the case that AD removes the step-size tuning and per-parameter cost of finite-difference DRCs while giving derivatives as accurate as an analytical solution, for both steady-state and transient kinetics. Cheap, exact local sensitivities also make sampling-based approaches such as DELSA practical for reaction networks with uncertain parameters. It is an early concrete example of the group’s argument that differentiable programming should be a routine tool in chemical engineering analysis (Yang et al. 2022).