Numerical solutions of partial differential equations (PDEs) require expensive simulations, limiting their application in design optimization routines, model-based control, or solution of large-scale inverse problems. Existing Convolutional Neural Network-based frameworks for surrogate modeling require lossy pixelization and data-preprocessing, which is not suitable for realistic engineering applications. Therefore, we propose non-linear independent dual system (NIDS), which is a deep learning surrogate model for discretizationindependent, continuous representation of PDE solutions, and can be used for prediction over domains with complex, variable geometries and mesh topologies. NIDS leverages implicit neural representations to develop a non-linear mapping between problem parameters and spatial coordinates to state predictions by combining evaluations of a case-wise parameter network and a point-wise spatial network in a linear output layer. The input features of the spatial network include physical coordinates augmented by a minimum distance function evaluation to implicitly encode the problem geometry. The form of the overall output layer induces a dual system, where each term in the map is non-linear and independent. Further, we propose a minimum distance function-driven weighted sum of NIDS models using a shared parameter network to enforce boundary conditions by construction under certain restrictions. The framework is applied to predict solutions around complex, parametrically-defined geometries on non-parametrically-defined meshes with solutions obtained many orders of magnitude faster than the full order models. Test cases include a vehicle aerodynamics problem with complex geometry and data scarcity, enabled by a training method in which more cases are gradually added as training progresses.
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