(298e) Manifold Learning Post-Processing Galerkin Algorithms for Dissipative PDEs on Their Approximate Inertial Manifolds

Dimensionality reduction to numerically solve partial differential equations (PDEs), involves the design of numerical algorithms that preserve the long-term properties of the original equation while improving the computational performance. Typically, the dimension is reduced to an attracting Inertial Manifold (AIMs), in which the PDE can be discretized as a finite system of ordinary differential equations. Galerkin methods computationally approximate the dynamics on approximate inertial manifolds (AIMs) through approximate inertial forms (AIFs). In the case of the Kuramoto-Sivashinsky (KS) equation, several nonlinear Galerkin schemes have been used to reduce the dimensionality while preserving dissipativity [1].

We propose a data driven nonlinear Galerkin scheme to approximate the AIFs of the KS. The method employs autoencoders and diffusion maps (based both on spectral and POD bases) to parametrize the reduced space. We then use neural networks and geometric harmonics to approximate the functional dependence between determining degrees of freedom and higher order ones. Finally, we reconstruct the full solution exploiting these dependencies, thus constructing a data driven postprocessing Galerkin scheme [2].

[1] Foias, C., Jolly, M. S., Kevrekidis, I. G., & Titi, E. S. (1994). On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation. Physics Letters A, 186(1-2), 87-96.

[2] Foias, C., Jolly, M. S., Kevrekidis, I. G., Sell, G. R., & Titi, E. S. (1988). On the computation of inertial manifolds. Physics Letters A, 131(7-8), 433-436.