Discretization-independent surrogate modeling over complex geometries using hypernetworks and implicit representations
This addresses the computational bottleneck in engineering applications like design optimization and inverse problems, offering a novel approach for realistic scenarios with variable geometry, though it is incremental in combining existing techniques like hypernetworks and implicit representations.
The paper tackled the problem of expensive numerical simulations for PDEs by proposing deep-learning surrogate models that provide discretization-independent, continuous representations of solutions over complex geometries, achieving predictions orders of magnitude faster than full-order models, with the best method showing competitive performance despite high parameter counts.
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.