# NTopo: Mesh-free Topology Optimization using Implicit Neural Representations

arXiv: https://arxiv.org/abs/2102.10782 · HTML: https://mlsanity.com/paper/2102.10782/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 67 references that are in the map.
- First posted on arXiv 22 Feb 2021.

## Builds on (most cited first, 23 of 23)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [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)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [PIFuHD: Multi-Level Pixel-Aligned Implicit Function for High-Resolution 3D Human Digitization](https://mlsanity.com/paper/2004.00452/) (2020)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [Neural Geometric Level of Detail: Real-time Rendering with Implicit 3D Shapes](https://mlsanity.com/paper/2101.10994/) (2021)
- [DiffTaichi: Differentiable Programming for Physical Simulation](https://mlsanity.com/paper/1910.00935/) (2019)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Learning to Control PDEs with Differentiable Physics](https://mlsanity.com/paper/2001.07457/) (2020)
- [ChainQueen: A Real-Time Differentiable Physical Simulator for Soft Robotics](https://mlsanity.com/paper/1810.01054/) (2018)
- [Physics informed deep learning for computational elastodynamics without labeled data](https://mlsanity.com/paper/2006.08472/) (2020)
- [TopologyGAN: Topology Optimization Using Generative Adversarial Networks Based on Physical Fields Over the Initial Domain](https://mlsanity.com/paper/2003.04685/) (2020)
- [NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation framework](https://mlsanity.com/paper/2012.07938/) (2020)
- [Neural reparameterization improves structural optimization](https://mlsanity.com/paper/1909.04240/) (2019)
- [Deep learning for determining a near-optimal topological design without any iteration](https://mlsanity.com/paper/1801.05463/) (2018)
- [3D Topology Optimization using Convolutional Neural Networks](https://mlsanity.com/paper/1808.07440/) (2018)
- [A deep Convolutional Neural Network for topology optimization with strong generalization ability](https://mlsanity.com/paper/1901.07761/) (2019)

## Built on by (most cited first, 30 of 41)

- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [CROM: Continuous Reduced-Order Modeling of PDEs Using Implicit Neural Representations](https://mlsanity.com/paper/2206.02607/) (2022)
- [Implicit Neural Spatial Representations for Time-dependent PDEs](https://mlsanity.com/paper/2210.00124/) (2022)
- [Simplicits: Mesh-Free, Geometry-Agnostic, Elastic Simulation](https://mlsanity.com/paper/2407.09497/) (2024)
- [Gradient-based Bi-level Optimization for Deep Learning: A Survey](https://mlsanity.com/paper/2207.11719/) (2022)
- [Neural Stress Fields for Reduced-order Elastoplasticity and Fracture](https://mlsanity.com/paper/2310.17790/) (2023)
- [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)
- [On Quantizing Implicit Neural Representations](https://mlsanity.com/paper/2209.01019/) (2022)
- [Geometry-Informed Neural Networks](https://mlsanity.com/paper/2402.14009/) (2024)
- [Neural topology optimization: the good, the bad, and the ugly](https://mlsanity.com/paper/2407.13954/) (2024)
- [Quaffure: Real-Time Quasi-Static Neural Hair Simulation](https://mlsanity.com/paper/2412.10061/) (2024)
- [Explorable INR: An Implicit Neural Representation for Ensemble Simulation Enabling Efficient Spatial and Parameter Exploration](https://mlsanity.com/paper/2504.00904/) (2025)
- [NeuralClothSim: Neural Deformation Fields Meet the Thin Shell Theory](https://mlsanity.com/paper/2308.12970/) (2023)
- [Differentiable Visual Computing for Inverse Problems and Machine Learning](https://mlsanity.com/paper/2312.04574/) (2023)
- [DMF-TONN: Direct Mesh-free Topology Optimization using Neural Networks](https://mlsanity.com/paper/2305.04107/) (2023)
- [Neural Metamaterial Networks for Nonlinear Material Design](https://mlsanity.com/paper/2309.10600/) (2023)
- [Simultaneous and Meshfree Topology Optimization with Physics-informed Gaussian Processes](https://mlsanity.com/paper/2408.03490/) (2024)
- [TopoGaussian: Inferring Internal Topology Structures from Visual Clues](https://mlsanity.com/paper/2503.12343/) (2025)
- [A Generalized Framework for Microstructural Optimization using Neural Networks](https://mlsanity.com/paper/2207.06512/) (2022)
- [Topology Optimization using Neural Networks with Conditioning Field Initialization for Improved Efficiency](https://mlsanity.com/paper/2305.10460/) (2023)
- [NITO: Neural Implicit Fields for Resolution-free Topology Optimization](https://mlsanity.com/paper/2402.05073/) (2024)
- [Neurally Integrated Finite Elements for Differentiable Elasticity on Evolving Domains](https://mlsanity.com/paper/2410.09417/) (2024)
- [Improving Efficiency of Iso-Surface Extraction on Implicit Neural Representations Using Uncertainty Propagation](https://mlsanity.com/paper/2402.13861/) (2024)
- [Compliance Minimization via Physics-Informed Gaussian Processes](https://mlsanity.com/paper/2507.09968/) (2025)
- [Edge Preserving Implicit Surface Representation of Point Clouds](https://mlsanity.com/paper/2301.04860/) (2023)
- [Multi-scale Topology Optimization using Neural Networks](https://mlsanity.com/paper/2404.08708/) (2024)
- [Regularized interpolation in 4D neural fields enables optimization of 3D printed geometries](https://mlsanity.com/paper/2502.01517/) (2025)
- [Diverse Topology Optimization using Modulated Neural Fields](https://mlsanity.com/paper/2502.13174/) (2025)

## Abstract (arXiv)

Recent advances in implicit neural representations show great promise when it comes to generating numerical solutions to partial differential equations. Compared to conventional alternatives, such representations employ parameterized neural networks to define, in a mesh-free manner, signals that are highly-detailed, continuous, and fully differentiable. In this work, we present a novel machine learning approach for topology optimization -- an important class of inverse problems with high-dimensional parameter spaces and highly nonlinear objective landscapes. To effectively leverage neural representations in the context of mesh-free topology optimization, we use multilayer perceptrons to parameterize both density and displacement fields. Our experiments indicate that our method is highly competitive for minimizing structural compliance objectives, and it enables self-supervised learning of continuous solution spaces for topology optimization problems.