Nonlinear Independent Dual System (NIDS) for Discretizationindependent Surrogate Modeling over Complex Geometries
Abstract
Numerical solutions of partial differential equations (PDEs) require expensive simulations, limiting their application in design optimization routines, modelbased control, or solution of largescale inverse problems. Existing Convolutional Neural Networkbased frameworks for surrogate modeling require lossy pixelization and datapreprocessing, which is not suitable for realistic engineering applications. Therefore, we propose nonlinear 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 nonlinear mapping between problem parameters and spatial coordinates to state predictions by combining evaluations of a casewise parameter network and a pointwise 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 nonlinear and independent. Further, we propose a minimum distance functiondriven 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, parametricallydefined geometries on nonparametricallydefined 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.07018
 Bibcode:
 2021arXiv210907018D
 Keywords:

 Physics  Computational Physics;
 Computer Science  Machine Learning