Geometry-Informed Neural Networks

* equal contribution
Main image
We train geometry-informed neural networks to produce shapes satisfying geometric design requirements. Under an additional diversity constraint, GINNs become generative. Not only produce multiple solutions but also learn a latent space structure. Traversing the 2D latent space continuously morphs solutions, i.e., the model generalizes. The latent space is also organized -- a central bulky shape becomes thinner in the radial direction and the axes can be identified by how the shape connects on the sides.
Image 1
The off-diagonal interpolates between anti-symmetric shapes.
Image 2
The main diagonal shifts mass along the back-front axis of the shape.

Abstract

Geometry is a ubiquitous tool in computer graphics, design, and engineering. However, the lack of large shape datasets limits the application of state-of-the-art supervised learning methods and motivates the exploration of alternative learning strategies. To this end, we introduce geometry-informed neural networks (GINNs) -- a framework for training shape-generative neural fields without data by leveraging user-specified design requirements in the form of objectives and constraints. By adding diversity as an explicit constraint, GINNs avoid mode-collapse and can generate multiple diverse solutions, often required in geometry tasks. Experimentally, we apply GINNs to several validation problems and a realistic 3D engineering design problem, showing control over geometrical and topological properties, such as surface smoothness or the number of holes. These results demonstrate the potential of training shape-generative models without data, paving the way for new generative design approaches without large datasets.

Validation examples

Min Surf
Minimal surface. GINN finds the unique surface that attaches to the prescribed boundary while having zero mean-curvature everywhere, also known as the Plateau's problem.
Mirror
Parabolic mirror. GINN finds the unique surface of a mirror -- a parabolic mirror -- that collects reflected rays into a single point.
Generative obstacle.
Obstacle. GINN finds the shape connecting the left and right interfaces within the allowed design region. Latent space traversal reveals a spectrum of solutions discovered via the diversity constraint.
Generative Physics-Informed neural network.
Under-determined physics. A generative physics-informed neural network (PINN) applied to an under-determined system of reaction-diffusion. Traversing the latent space of the trained network produces morphing Turing patterns.

Future work

GINNs are a work in-progress and there are many future directions to explore:
  • including data,
  • including physics through PDE constraints,
  • different diversity measures,
  • speeding up the training,
  • second-order optimizers,
  • scaling the experiments,
  • neural field conditioning mechanisms.

BibTeX

@article{berzins2024ginn,
  title={Geometry-Informed Neural Networks},
  author={Berzins, Arturs and Radler, Andreas and Volkmann, Eric and Sanokowski, Sebastian and Hochreiter, Sepp, and Brandstetter, Johannes},
  journal={arXiv preprint arXiv:2402.14009},
  year={2024}
}

Acknowledgements

This project was supported by the European Union’s Horizon 2020 Research and Innovation Programme under Grant Agreement number 860843.