Fig. 2.

Neural network training and performance. We generated 10⁵ simulated spatial distributions using our partial differential equation (PDE) model and split the data into three groups: 80% for training, 10% for validation and 10% for test. We used root mean squared errors (RMSEs) to evaluate the differences between data generated by the mechanism-based model and data generated by the neural network. a Accuracy of the trained neural network. The top panel shows the predicted distributions by the neural network plotted against the distributions generated by numerical simulations. The bottom panel shows the peak values predicted by the neural network plotted against the peak values generated by numerical simulations. Perfect alignment corresponds to the y = x line. The test sample size is s (=10,000). Each spatial distribution consists of 501 discrete points; thus, the top panel consists of 5,010,000 points. b Representative distributions predicted by neural network from test dataset. Each blue line represents a predicted distribution using the trained neural network; the corresponding red dashed line represents that generated by a numerical simulation. Additional examples are shown in Supplementary Figs. 3 and 4. c Identifying the appropriate data size for reliable training. The top panel shows the RMSE between distributions generated by the neural network and the distributions generated by numerical simulations as a function of an increasing training data size. The bottom panel shows the RMSE of peak-value predictions as a function of an increasing training data size. The RMSEs are calculated based on predictions of a test dataset, which contains 20,000 samples