Speaker
Description
Deep learning architectures have recently been investigated for fast prediction in parametric PDEs. Within the Model Order Reduction (MOR) paradigm, an offline stage projects the nonlinear solution space into a low-dimensional manifold, and the compressed latent representation enables rapid, accurate predictions with a small computational cost.
In particular, Graph Neural Networks have shown high adaptability to exploit topological informations in modeling high-dimensional dynamical systems governed by partial differential equations on irregular meshes.
We propose a geometry-informed surrogate for nonlinear flow dynamics in which the metric and manifold structure are encoded via a Laplacian eigenvectors embedding, while discrete topological representations are preserved by permutation invariant message-passing on undirected graphs. We also use the Laplacian basis to perform a Graph Fourier transform, enabling spectral filtering of node features.
Dimensionality is reduced through error-guided supernodes selection, to aggregate informations towards high-error regions.
A multiscale, geometry-aware graph autoencoder is trained to learn a compact latent solution manifold. We then integrate the latent dynamics with a Neural ODE, yielding stable long-horizon rollouts and improved generalization across domains and Reynolds regimes.
We perform Data Assimilation directly in the latent space using a Deterministic Ensemble Kalman Filter update. The observation operator is the decoder, so corrections are geometrically consistent with the original solution manifold.
Preliminary experiments on 2D unsteady flows with $Re\in [100,1000]$ indicate that the Laplacian eigenvector features reduce the amount of training data needed to achieve smooth, geometry-consistent reconstructions; the latent space continuous dynamics provides long-horizon stability, reducing error accumulation, and the EnKF updates perform online data assimilation, using sparse sensor observations with quantified uncertainty to correct residual drift.