# PCNN: A physics-constrained neural network for multiphase flows

arXiv: https://arxiv.org/abs/2109.08965 · HTML: https://mlsanity.com/paper/2109.08965/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 77 references that are in the map.
- First posted on arXiv 18 Sep 2021.

## Builds on (most cited first, 24 of 24)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Bidirectional LSTM-CRF Models for Sequence Tagging](https://mlsanity.com/paper/1508.01991/) (2015)
- [Conditional Neural Processes](https://mlsanity.com/paper/1807.01613/) (2018)
- [LSTM: A Search Space Odyssey](https://mlsanity.com/paper/1503.04069/) (2015)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Physics-guided Neural Networks (PGNN): An Application in Lake Temperature Modeling](https://mlsanity.com/paper/1710.11431/) (2017)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [Predicting Human Eye Fixations via an LSTM-based Saliency Attentive Model](https://mlsanity.com/paper/1611.09571/) (2016)
- [Deep Learning of Vortex Induced Vibrations](https://mlsanity.com/paper/1808.08952/) (2018)
- [Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equations](https://mlsanity.com/paper/1907.03507/) (2019)
- [Deep convolutional encoder-decoder networks for uncertainty quantification of dynamic multiphase flow in heterogeneous media](https://mlsanity.com/paper/1807.00882/) (2018)
- [Physics Guided RNNs for Modeling Dynamical Systems: A Case Study in Simulating Lake Temperature Profiles](https://mlsanity.com/paper/1810.13075/) (2018)
- [Learning Scalable Deep Kernels with Recurrent Structure](https://mlsanity.com/paper/1610.08936/) (2016)
- [Deep Theory of Functional Connections: A New Method for Estimating the Solutions of PDEs](https://mlsanity.com/paper/1812.08625/) (2018)
- [Highly-scalable, physics-informed GANs for learning solutions of stochastic PDEs](https://mlsanity.com/paper/1910.13444/) (2019)
- [Reduced order modeling of subsurface multiphase flow models using deep residual recurrent neural networks](https://mlsanity.com/paper/1810.10422/) (2018)

## Built on by (most cited first, 1 of 1)

- [CEENs: Causality-enforced evolutional networks for solving time-dependent partial differential equations](https://mlsanity.com/paper/2610.04405/) (2026)

## Abstract (arXiv)

The present study develops a physics-constrained neural network (PCNN) to predict sequential patterns and motions of multiphase flows (MPFs), which includes strong interactions among various fluid phases. To predict the order parameters, which locate individual phases in the future time, a neural network (NN) is applied to quickly infer the dynamics of the phases by encoding observations. The multiphase consistent and conservative boundedness mapping algorithm (MCBOM) is next implemented to correct the predicted order parameters. This enforces the predicted order parameters to strictly satisfy the mass conservation, the summation of the volume fractions of the phases to be unity, the consistency of reduction, and the boundedness of the order parameters. Then, the density of the fluid mixture is updated from the corrected order parameters. Finally, the velocity in the future time is predicted by another NN with the same network structure, but the conservation of momentum is included in the loss function to shrink the parameter space. The proposed PCNN for MPFs sequentially performs (NN)-(MCBOM)-(NN), which avoids nonphysical behaviors of the order parameters, accelerates the convergence, and requires fewer data to make predictions. Numerical experiments demonstrate that the proposed PCNN is capable of predicting MPFs effectively.