where N denotes the hidden dynamics as a non-linear function of the emergent space, ζ, emergent time τ , the solution, Ĉ, and its spatial derivatives, Ĉζ , Ĉζ ζ , as well as higher-order derivatives. We assume here that the system has only a first-order time derivative of the solution, Ĉτ , in its governing equation. DHPM involves a sequential integration of two DNNs: a solution network, N sol , and a hidden physics network, N hid . N sol approximates the system state, Ĉ( ζ, τ ), while N hid captures the latent dynamics encoded by the governing physical equations, specifically the part denoted by N in . By jointly training these networks using a composite loss function, DHPMs seek to minimize the discrepancy between observed data and model predictions while enforcing consistency with underlying physics. The composite loss function, L comprises two components: a data loss term, L D , and a physics loss term, L P DE . The data loss term quantifies the deviation between model predictions and observed data (equivalent to Equation ), whereas the physics loss term conventionally written in terms of residual loss obtained from Equation ( ), constrains the model to adhere to known physical laws or equations. Similar to the physics loss of Equation ( ), in DHPMs we define the physics loss over N c collocation points as follows: