# Surrogate Modeling for Fluid Flows Based on Physics-Constrained Deep Learning Without Simulation Data

## TL;DR

This paper introduces a physics-constrained deep learning approach to create surrogate models for fluid flows that do not require any simulation data, enabling efficient and accurate predictions in complex fluid dynamics problems.

## Contribution

It proposes a novel structured deep neural network architecture that enforces physical laws and boundary conditions without relying on simulation data, advancing surrogate modeling techniques.

## Key findings

- Accurately predicts fluid flow fields without simulation data.
- Effectively propagates uncertainties in fluid properties and geometry.
- Achieves excellent agreement with first-principle numerical simulations.

## Abstract

Numerical simulations on fluid dynamics problems primarily rely on spatially or/and temporally discretization of the governing equation into the finite-dimensional algebraic system solved by computers. Due to complicated nature of the physics and geometry, such process can be computational prohibitive for most real-time applications and many-query analyses. Therefore, developing a cost-effective surrogate model is of great practical significance. Deep learning (DL) has shown new promises for surrogate modeling due to its capability of handling strong nonlinearity and high dimensionality. However, the off-the-shelf DL architectures fail to operate when the data becomes sparse. Unfortunately, data is often insufficient in most parametric fluid dynamics problems since each data point in the parameter space requires an expensive numerical simulation based on the first principle, e.g., Naiver--Stokes equations. In this paper, we provide a physics-constrained DL approach for surrogate modeling of fluid flows without relying on any simulation data. Specifically, a structured deep neural network (DNN) architecture is devised to enforce the initial and boundary conditions, and the governing partial differential equations are incorporated into the loss of the DNN to drive the training. Numerical experiments are conducted on a number of internal flows relevant to hemodynamics applications, and the forward propagation of uncertainties in fluid properties and domain geometry is studied as well. The results show excellent agreement on the flow field and forward-propagated uncertainties between the DL surrogate approximations and the first-principle numerical simulations.