Solving parametric PDEs on solution manifolds parametrized by neural networks

A slow decaying Kolmogorov n-width of the solution manifold of a parametric partial differential equation precludes the realization of efficient linear projection-based reduced-order models. This is due to the high dimensionality of the reduced space needed to approximate with sufficient accuracy the solution manifold. To solve this problem, neural networks, in the form of different architectures, have been employed to build accurate non-linear regressions of the solution manifolds. However, the majority of the implementations are non-intrusive black-box surrogate models.

Left: evolution trajectories on the r-dimensional latent space and non-linear solution manifold in the ambient space R^d. The map φ is a single chart non-linear parametrization of the approximating solution manifold. Right: evolution trajectories on the r-dimensional linear latent space and solution manifold embedded in the ambient space R^d. The map Pr is a linear projection.

Recently [1], a new explicable methodology for reduced-order modelling has been developed. This methodology employs neural networks for solution manifold approximation but does not discard the physical and numerical models underneath during the predictive/online stage.

The focus is on auto-encoders used to further compress the dimensionality of linear approximants of solution manifolds, ultimately achieving non-linear dimension reduction. After obtaining an accurate non-linear approximant, the solutions on the latent manifold are sought using the residual-based non-linear least-squares Petrov-Galerkin method [3], appropriately hyper-reduced to be independent of the number of degrees of freedom.

Hyper-reduced architecture employed. Left: the decoder map followed by the vector matrix multiplication with the randomized singular value decomposition modes . The latter are restricted to the sub-mesh through the projection: the blackened degrees of freedom are discarded. Right: the map actually employed in the hyper-reduced non-linear manifold least-squares Petrov-Galerkin method. It is independent of the number of degrees of freedom and its evaluation is efficient thanks to the relatively small size of the decoder.

Above is displayed the hyper-reduced decoder map employed:

Restriction of the encoder map. Left: original decoder map from the latent space of dimension r<<d to the d-dimensional state space. Right: employed decoder map from the latent space of dimension r<<d to the s-dimensional sub-mesh.

In which we have assumed that the discretized solutions of the parametric PDE under study, belong to

A compression f_filter and reconstruction map from reduced to full-state space is defined. In our case, we employed parallel randomized singular value decomposition.

Reconstruction map: from the space of p-dimensional reduced coordinates to the d-dimensional space of state variables.

A key ingredient is the projection map into the sub-mesh prescribed by the chosen hyper-reduction method:

Restriction of the d-dimensional state variables to the sub-mesh of dimension s.

Adaptive and local hyper-reduction strategies can be developed along with the employment of local non-linear approximants.

As testcase [1], we consider the incompressible turbulent flow around the Ahmed body [2]. The parameters are time and the slant angle of Ahmed’s body.

Computational domain

Lateral slice of the computational domain. Left: predicted velocity field. Right: adaptive hyper-reduction sub-mesh.

Bottom slice of the computational domain. Left: predicted pressure field. Right: adaptive hyper-reduction sub-mesh.

References:

[1] Romor, Francesco, Giovanni Stabile, and Gianluigi Rozza. “Explicable hyper-reduced order models on nonlinearly approximated solution manifolds of compressible and incompressible Navier-Stokes equations.” arXiv preprint arXiv:2308.03396 (2023).

[2] Zancanaro, Matteo, et al. “Hybrid neural network reduced order modelling for turbulent flows with geometric parameters.” Fluids 6.8 (2021): 296.

[3] Romor, Francesco, Giovanni Stabile, and Gianluigi Rozza. “Non-linear manifold reduced-order models with convolutional autoencoders and reduced over-collocation method.” Journal of Scientific Computing 94.3 (2023): 74.