Discovering Bäcklund Transformations with PDE Foundation Models

We propose a novel application of Foundation Models trained on multi–Partial-Differential-Equation data. Leveraging the vector embeddings learnt by one such model, we discuss a necessary condition for the existence of a Bäcklund transformation between any pairs of Partial Differential Equations in its training dataset, which can be used when certain requirements on the dimension of the embedding space and the size of the training datasets are satisfied. In this case, the condition assumes a simple linear form and its computation scales no faster than O((MN)^3).