# Detecting hidden structures from a static loading experiment: topology optimization meets physics-informed neural networks

arXiv: https://arxiv.org/abs/2303.09280 · HTML: https://mlsanity.com/paper/2303.09280/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 77 references that are in the map.
- First posted on arXiv 13 Mar 2023.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [On Hyperparameter Optimization of Machine Learning Algorithms: Theory and Practice](https://mlsanity.com/paper/2007.15745/) (2020)
- [Gradient Descent with Early Stopping is Provably Robust to Label Noise for Overparameterized Neural Networks](https://mlsanity.com/paper/1903.11680/) (2019)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks](https://mlsanity.com/paper/2007.07442/) (2020)
- [Physics informed deep learning for computational elastodynamics without labeled data](https://mlsanity.com/paper/2006.08472/) (2020)
- [Learning in Sinusoidal Spaces with Physics-Informed Neural Networks](https://mlsanity.com/paper/2109.09338/) (2021)
- [Optimal control of PDEs using physics-informed neural networks](https://mlsanity.com/paper/2111.09880/) (2021)
- [Physics informed neural networks for continuum micromechanics](https://mlsanity.com/paper/2110.07374/) (2021)
- [A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method](https://mlsanity.com/paper/2206.13103/) (2022)
- [Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domains](https://mlsanity.com/paper/2302.04954/) (2023)
- [Using explainability to design physics-aware CNNs for solving subsurface inverse problems](https://mlsanity.com/paper/2211.08651/) (2022)
- [Optimal Design of Responsive Structures](https://mlsanity.com/paper/2107.08895/) (2021)

## Abstract (arXiv)

Most noninvasive imaging techniques utilize electromagnetic or acoustic waves originating from multiple locations and directions to identify hidden geometrical structures. Surprisingly, it is also possible to image hidden voids and inclusions buried within an object using a single static thermal or mechanical loading experiment by observing the response of the exposed surface of the body, but this problem is challenging to invert. Although physics-informed neural networks (PINNs) have shown promise as a simple-yet-powerful tool for problem inversion, they have not yet been applied to imaging problems with a priori unknown topology. Here, we introduce a topology optimization framework based on PINNs that identifies concealed geometries using exposed surface data from a single loading experiment, without prior knowledge of the number or types of shapes. We allow for arbitrary solution topology by representing the geometry using a material density field combined with a novel eikonal regularization technique. We validate our framework by detecting the number, locations, and shapes of hidden voids and inclusions in many example cases, in both 2D and 3D, and we demonstrate the method's robustness to noise and sparsity in the data. Our methodology opens a pathway for PINNs to solve geometry optimization problems in engineering.