In the DHPM, we focused on a single experiment which was defined for a single Péclet number. For real-time inference, training a DHPM for each experiment may not be feasible. To that end, we aim to construct an efficient emulator to generalize related scenarios and have real-time inference. In this example, we construct a DeepONet on the emergent space to map π, ζ, and τ to the solution on the emergent space. Mathematically, we are learning an operator Ψ parameterized by Θ such that for Θ = θ * the DeepONet accurately approximates the desired mapping. This is represented as