DNN-Based Topology Optimisation: Spatial Invariance and Neural Tangent Kernel
This work addresses a specific issue in computational design and engineering for topology optimization, offering an incremental improvement to existing neural network-based methods.
The paper tackled the problem of visual artifacts and suboptimal shapes in topology optimization using neural networks by showing that the Neural Tangent Kernel (NTK) filter is not translation-invariant, and they proposed two input embeddings to achieve approximate spatial invariance, empirically confirming improved results.
We study the Solid Isotropic Material Penalisation (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.