where Ψ θ (π 1 )( ζj , τj ) is the prediction of the DeepONet and Ψ(π 1 )( ζj , τj ) is the ground truth of the solution defined in the emergent space using the questionnaire algorithm. The branch and trunk networks can be modeled with any DNN architecture. Here, we consider a fully connected feed-forward NN for both networks (branch and trunk). A schematic representation of the DeepONet architecture for the advection-diffusion problem (winding pipe example) is shown in Figure (b).