DNN-Based Topology Optimisation: Spatial Invariance and Neural Tangent Kernel
arXiv:2106.05710
Abstract
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.
References in corpus (11)
- Neural Tangent Kernel: Convergence and Generalization in Neural Networks
- Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains
- On Exact Computation with an Infinitely Wide Neural Net
- Exponential expressivity in deep neural networks through transient chaos
- Toward Deeper Understanding of Neural Networks: The Power of Initialization and a Dual View on Expressivity
- Neural reparameterization improves structural optimization
- 3D Topology Optimization using Convolutional Neural Networks
- Dynamics of Deep Neural Networks and Neural Tangent Hierarchy
- A New CGAN Technique for Constrained Topology Design Optimization
- Order and Chaos: NTK views on DNN Normalization, Checkerboard and Boundary Artifacts
- Length Scale Control in Topology Optimization using Fourier Enhanced Neural Networks