Efficient parameterization of invariant manifolds using deep neural networks

Friday, September 23, 2022 - 11:00am for 1 hour (actually 50 minutes)
Shane Kepley – VU – s.kepley@vu.nl
Jorge Gonzalez


Spectral methods are the gold standard for parameterizing manifolds of solutions for ODEs because of their high precision and amenability to computer assisted proofs. However, these methods suffer from several drawbacks. In particular, the parameterizations are costly to compute and time-stepping is far more complicated than other methods. In this talk we demonstrate how computing these parameterizations and accurately time-stepping can be reduced to a related manifold learning problem. The latter problem is solved by training a deep neural network to interpolate charts for a low dimensional manifold embedded in a high dimensional Euclidean space. This training is highly parallelizable and need only be performed once. Once the neural network is trained, it is capable of parameterizing invariant manifolds for the ODE and time-stepping with remarkable efficiency and precision.