Today I had a great meeting with Wilson Gregory (JHU), Drummond Fielding (Flatiron), and Soledad Villar (JHU) about a project to learn partial differential equations from simulation data. This is a toy problem from our perspective, but it is a baby step towards big, real computational problems in physics. Fielding produces training data, Gregory produces geometric convolution networks, Villar proves things, and I cheer from the sidelines.
Our approach is to replace convolutional neural networks with geometric operators that are generalizations of convolutions that know more about geometric forms like vectors, tensors, pseudovectors, and so on. By building methods that use geometric objects responsibly, we automatically enforce coordinate freedom and other deep symmetries.
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