This talk introduces new ideas for learning reliable representations by explicitly preserving geometric structure. I'll present recent work in which we treat data as fields with geometric latent variables, modeled through equivariant neural fields [1]. The equivariance relation ensures that the latent variables are geometrically meaningful, which we show through various classification and segmentation experiments. I subsequently show that this allows for continuous PDE forecasting entirely in latent space [2]. That is, instead of modeling a PDE on the fields, we model equivariant dynamics through equivariant neural ODEs applied to the geometric latent variables. I'll further highlight a generalized notion of weight-sharing as sharing operations over equivalence classes of point-pairs [3]. Through this principle, we developed a general-purpose equivariant neural network that achieves state-of-the-art performance on a wide range of tasks, from regression to segmentation and the generation of novel molecules. I conclude with our most recent work on designable (and thus interpretable) latent space through isometry learning and pull-back geometry [4].

 

[1] Wessels, D. R., Knigge, D. M., Papa, S., Valperga, R., Vadgama, S., Gavves, E., & Bekkers, E. J. (2024). Grounding Continuous Representations in Geometry: Equivariant Neural Fields. arXiv preprint arXiv:2406.05753. ISO 690

[2] [accepted at Neurips 2024] Knigge, D. M., Wessels, D. R., Valperga, R., Papa, S., Sonke, J. J., Gavves, E., & Bekkers, E. J. (2024). Space-Time Continuous PDE Forecasting using Equivariant Neural Fields. arXiv preprint arXiv:2406.06660.

[3] Bekkers, E. J., Vadgama, S., Hesselink, R., Van der Linden, P. A., & Romero, D. W. Fast, Expressive $\mathrm {SE}(n) $ Equivariant Networks through Weight-Sharing in Position-Orientation Space. In The Twelfth International Conference on Learning Representations. ISO 690

[4] de Kruiff, F., Bekkers, E., Öktem, O., Schönlieb, C. B., & Diepeveen, W. (2024). Pullback Flow Matching on Data Manifolds. arXiv preprint arXiv:2410.04543