# DNN-Based Topology Optimisation: Spatial Invariance and Neural Tangent Kernel

arXiv: https://arxiv.org/abs/2106.05710 · HTML: https://mlsanity.com/paper/2106.05710/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 35 references that are in the map.
- First posted on arXiv 10 Jun 2021.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Exponential expressivity in deep neural networks through transient chaos](https://mlsanity.com/paper/1606.05340/) (2016)
- [Deep Information Propagation](https://mlsanity.com/paper/1611.01232/) (2016)
- [Toward Deeper Understanding of Neural Networks: The Power of Initialization and a Dual View on Expressivity](https://mlsanity.com/paper/1602.05897/) (2016)
- [Dynamics of Deep Neural Networks and Neural Tangent Hierarchy](https://mlsanity.com/paper/1909.08156/) (2019)
- [TopologyGAN: Topology Optimization Using Generative Adversarial Networks Based on Physical Fields Over the Initial Domain](https://mlsanity.com/paper/2003.04685/) (2020)
- [Neural reparameterization improves structural optimization](https://mlsanity.com/paper/1909.04240/) (2019)
- [Neural networks for topology optimization](https://mlsanity.com/paper/1709.09578/) (2017)
- [Global optimization of dielectric metasurfaces using a physics-driven neural network](https://mlsanity.com/paper/1906.04157/) (2019)
- [3D Topology Optimization using Convolutional Neural Networks](https://mlsanity.com/paper/1808.07440/) (2018)
- [Disentangling Trainability and Generalization in Deep Neural Networks](https://mlsanity.com/paper/1912.13053/) (2019)
- [A New CGAN Technique for Constrained Topology Design Optimization](https://mlsanity.com/paper/1901.07675/) (2019)

## Built on by (most cited first, 2 of 2)

- [Neural Tangent Kernel: A Survey](https://mlsanity.com/paper/2208.13614/) (2022)
- [Neural topology optimization: the good, the bad, and the ugly](https://mlsanity.com/paper/2407.13954/) (2024)

## Abstract (arXiv)

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.