# Quantifying implicit and explicit constraints on physics-informed neural processes

Due to strong interactions among various phases and among the phases and fluid motions, multiphase flows (MPFs) are so complex that lots of efforts have to be paid to predict its sequential patterns of phases and motions. The present paper applies the physical constraints inherent in MPFs and enforces them to a physics-informed neural network (PINN) model either explicitly or implicitly, depending on the type of constraints. To predict the unobserved order parameters (OPs) (which locate the phases) in the future steps, the conditional neural processes (CNPs) with long short-term memory (LSTM, combined as CNPLSTM) are applied to quickly infer the dynamics of the phases after encoding only a few observations. After that, the multiphase consistent and conservative boundedness mapping (MCBOM) algorithm is implemented the correction the predicted OPs from CNP-LSTM so that the mass conservation, the summation of the volume fractions of the phases being unity, the consistency of reduction, and the boundedness of the OPs are strictly satisfied. Next, the density of the fluid mixture is computed from the corrected OPs. The observed velocity and density of the fluid mixture then encode in a physics-informed conditional neural processes and long short-term memory (PICNP-LSTM) where the constraint of momentum conservation is included in the loss function. Finally, the unobserved velocity in future steps is predicted from PICNP-LSTM. The proposed physics-informed neural processes (PINPs) model (CNP-LSTM-MCBOM-PICNP-LSTM) for MPFs avoids unphysical behaviors of the OPs, accelerates the convergence, and requires fewer data. The proposed model successfully predicts several canonical MPF problems, i.e., the horizontal shear layer (HSL) and dam break (DB) problems, and its performances are validated.