Discovering Bäcklund Transformations with PDE Foundation Models

Eloisa Bentivegna, Tong Luo

Keywords: Partial Differential Equations, Foundation Models, Bäcklund Transformations

TL;DR: We present a novel use of PDE Foundation Models, and derive a condition for its applicability.

Abstract: 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 $M$ and the size of the training datasets $N$ are satisfied. In this case, the condition assumes a simple linear form and its computation scales no faster than $O((MN)^3)$.

Submission Number: 129