# Length Scale Control in Topology Optimization using Fourier Enhanced Neural Networks

arXiv: https://arxiv.org/abs/2109.01861 · HTML: https://mlsanity.com/paper/2109.01861/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Sep 2021.

## Built on by (most cited first, 11 of 11)

- [Topology Optimization via Machine Learning and Deep Learning: A Review](https://mlsanity.com/paper/2210.10782/) (2022)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [DMF-TONN: Direct Mesh-free Topology Optimization using Neural Networks](https://mlsanity.com/paper/2305.04107/) (2023)
- [Concurrent build direction, part segmentation, and topology optimization for additive manufacturing using neural networks](https://mlsanity.com/paper/2210.01315/) (2022)
- [Topology Optimization using Neural Networks with Conditioning Field Initialization for Improved Efficiency](https://mlsanity.com/paper/2305.10460/) (2023)
- [PATO: Producibility-Aware Topology Optimization using Deep Learning for Metal Additive Manufacturing](https://mlsanity.com/paper/2112.04552/) (2021)
- [Gradient-free neural topology optimization: Towards effective fracture-resistant designs](https://mlsanity.com/paper/2403.04937/) (2024)
- [Multi-scale Topology Optimization using Neural Networks](https://mlsanity.com/paper/2404.08708/) (2024)
- [Heterogeneous Metamaterials Design via Multiscale Neural Implicit Representation](https://mlsanity.com/paper/2511.03012/) (2025)
- [TopoEdit: Fast Post-Optimization Editing of Topology Optimized Structures](https://mlsanity.com/paper/2602.22430/) (2026)
- [How the Classical Baseline Shapes Machine-Learning Speed Claims in Topology Optimization](https://mlsanity.com/paper/2610.16051/) (2026)

## Abstract (arXiv)

Length scale control is imposed in topology optimization (TO) to make designs amenable to manufacturing and other functional requirements. Broadly, there are two types of length-scale control in TO: emph exact and emph approximate. While the former is desirable, its implementation can be difficult, and is computationally expensive. Approximate length scale control is therefore preferred, and is often sufficient for early stages of design. In this paper we propose an approximate length scale control strategy for TO, by extending a recently proposed density-based TO formulation using neural networks (TOuNN). Specifically, we enhance TOuNN with a Fourier space projection, to control the minimum and/or maximum length scales. The proposed method does not involve additional constraints, and the sensitivity computations are automated by expressing the computations in an end-end differentiable fashion using the neural net's library. The proposed method is illustrated through several numerical experiments for single and multi-material designs.