Fig. 2: Simulation results of solving Darcy flow and magnetostatic Poisson’s equations.

a Illustration of the Darcy flow equation describing a fluid flow through a porous medium. The optical neural engine (ONE) architecture learns the mapping between the permeability and pressure fields. b Training loss curves for input data with the input data resolutions of 85, 106, 141, 211, and 421. c Comparison of the training loss of fully convolution networks (FCN), principal component analysis-based neural network (PCANN), reduced biased method (RBM), graph neural operator (GNO), low-rank kernel decomposition neural operator (LNO), multipole graph neural operator (MGNO), Fourier neural operator (FNO), and ONE models at various resolutions. d Input permeability field, the expected ground truth of output pressure field, the predicted output pressure field, the absolute error between the expected and predicted outputs, and the relative error between the expected and predicted outputs, at 85 and 421 resolutions. e Illustration of the magnetostatic Poisson’s equation calculating the demagnetizing field generated by the magnetization field. The ONE architecture learns the mapping between these two fields. f Validation loss curve for the ONE architecture solving the magnetostatic Poisson’s equation and (g) corresponding input magnetization field, the expected ground truth of output demagnetizing field, the predicted output demagnetizing field, the absolute and normalized errors between the expected and predicted outputs.