  Numerical solutions of partial differential equations (PDEs) require
expensive simulations, limiting their application in design optimization,
model-based control, and large-scale inverse problems. Surrogate modeling
techniques seek to decrease the computational expense while retaining dominant
solution features and behavior. Traditional Convolutional Neural Network-based
frameworks for surrogate modeling require lossy pixelization and
data-preprocessing, and generally are not effective in realistic engineering
applications. We propose alternative deep-learning based surrogate models for
discretization-independent, continuous representations of PDE solutions, which
can be used for learning and prediction over domains with complex, variable
geometry and mesh topology. Three methods are proposed and compared;
design-variable-coded multi-layer perceptron (DV-MLP), design-variable
hypernetworks (DV-Hnet), and non-linear independent dual system (NIDS). Each
method utilizes a main network which consumes pointwise spatial information to
provide a continuous representation, allowing predictions at any location in
the domain. Input features include a minimum-distance function evaluation to
implicitly encode the problem geometry. The geometric design variables, which
define and distinguish problem instances, are used differently by each method,
appearing as additional main-network input features (DV-MLP), or as
hypernetwork inputs (DV-Hnet and NIDS). The methods are applied to predict
solutions around complex, parametrically-defined geometries on
non-parametrically-defined meshes with model predictions obtained many orders
of magnitude faster than the full order models. Test cases include a
vehicle-aerodynamics problem with complex geometry and limited training data,
with a design-variable hypernetwork performing best, with a competitive
time-to-best-model despite a much greater parameter count.
