Rotationally Equivariant Neural Operators for Learning Transformations on Tensor Fields (eg 3D Images and Vector Fields)

We introduce equivariant neural operators for learning resolution invariant as well as translation and rotation equivariant transformations between sets of tensor fields. Input and output may contain arbitrary mixes of scalar fields, vector fields, second order tensor fields and higher order fields. Our tensor field convolution layers emulate any linear operator by learning its impulse response or Green's function as the convolution kernel. Our tensor field attention layers emulate pairwise field coupling via local tensor products. Convolutions and associated adjoints can be in real or Fourier space allowing for linear scaling. By unifying concepts from E3NN, TBNN and FNO, we achieve good predictive performance on a wide range of PDEs and dynamical systems in engineering and quantum chemistry. Code is in Julia and available upon request from authors.

Computing Consensus Curves

We consider the problem of extracting accurate average ant trajectories from many (possibly inaccurate) input trajectories contributed by citizen scientists. Although there are many generic software tools for motion tracking and specific ones for insect tracking, even untrained humans are much better at this task, provided a robust method to computing the average trajectories. We implemented and tested several local (one ant at a time) and global (all ants together) method. Our best performing algorithm uses a novel global method, based on finding edge-disjoint paths in an ant-interaction graph constructed from the input trajectories. The underlying optimization problem is a new and interesting variant of network flow. Even though the problem is NP-hard, we implemented two heuristics, which work very well in practice, outperforming all other approaches, including the best automated system.