where [b 1 , b 2 , . . . , b q ] T is the output vector of the branch net, [tr 1 , tr 2 , . . . , tr q ] T the output vector of the trunk net and q denotes a hyperparameter that controls the size of the output layer of both the branch and trunk networks. The trainable parameters of the DeepONet, represented by θ in Equation ( ), are obtained by minimizing a data-driven loss function, which is expressed as: