DNN-based Topology Optimisation: Spatial Invariance and Neural Tangent Kernel

Benjamin Dupuis, Arthur Jacot

Keywords: Neural Tangent Kernel, Topology optimization, DNN, Neural Networks, positional encoding

TL;DR: Theoretical analysis of topological optimization with DNNs using the NTK.

Abstract: We study the Solid Isotropic Material Penalization (SIMP) method with a density field generated by a fully-connected neural network, taking the coordinates as inputs. In the large width limit, we show that the use of DNNs leads to a filtering effect similar to traditional filtering techniques for SIMP, with a filter described by the Neural Tangent Kernel (NTK). This filter is however not invariant under translation, leading to visual artifacts and non-optimal shapes. We propose two embeddings of the input coordinates, which lead to (approximate) spatial invariance of the NTK and of the filter. We empirically confirm our theoretical observations and study how the filter size is affected by the architecture of the network. Our solution can easily be applied to any other coordinates-based generation method.

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Code: https://github.com/benjiDupuis/DeepTopo