Fig. 3: Comparison of Single- and Multi-Start Optimisation for UDE model of Glycolysis.

a The UDE framework is applied in three stages to the Glycolysis model. The definition of the UDE problem integrates interpretable components derived from prior knowledge (shown in green) with a flexible term—potentially capturing ATP usage—represented by an ANN (shown in red). The figure displays 2 of the 7 differential equations governing the state variables; the full system is provided in the section “Problem scenarios”. b–d Best fit on training data and corresponding prediction achieved by standard single-, adapted single-, and multi-start approaches for the data set with 46 data points and 5% noise. The standard deviation was estimated jointly with the other parameters. Note that this is only possible for the approaches using the NLL as objective function, i.e. the adapted single start and multi-start pipeline. e Comparison of the NMAE on the training and test set by approach. The test loss for the adapted approach is not computable due to the simulation failure (c). f The success rate (Test loss < 0.15) for UDEs in the multi-start pipeline. g Waterfall plot of the training loss for the 100 best optimisation runs of the multi-start approach.