Fig. 7. ANN-aided model approximation seamlessly integrates the inference of kinetic parameters and approximation of the delay CME by a NN-CME.

The unknown kinetic parameters can be treated in the same way as neural-network coefficients (weight and biases) and optimized to minimize the objective function. Application to Model II. a Sketch of the computation of the 95% confidence interval (CI) of the inferred kinetic parameters. Blue areas indicate the 95% confidence region, while the grey area shows the non-confidence region. Both solid and dashed red lines show the profile likelihoods (PLs) of burst frequency α and burst size b, respectively (See SI Note 8 for details). b Inferred values of α and burst size b (dots), their 95% CIs (error bars) and the true values (green lines) for five mammalian genes. Inference by using ANN-aided model approximation is robust against size of dataset: Dataset A (blue, 100 snapshots and 10⁴ cells) and Dataset B (red, 50 snapshots and 10³ cells) produce similar results. c Quantile–quantile plots for the steady-state distributions of the NN-CME and those obtained from the SSA; the linearity confirms that the ANN-aided model approximation can accurately approximate the distribution using the NN-CME even when the optimization is over both the kinetic parameters and the neural-network coefficients. The rate constants and other parameters related to the ANN’s training are specified in SI Table 1.