  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.
