Reduced Order Modeling

Reduced order modeling is a data-driven technique that exploits the structure of parametric partial differential equations (PPDEs) to make repeated simulations of this PPDE much cheaper.

For this consider a PPDE written in the form: $F(z(\mu);\mu)=0$ where $z(\mu)$ evolves on a infinite-dimensional Hilbert space $V$.

In modeling any PDE we have to choose a discretization (particle discretization, finite element method, ...) of $V$, which will be denoted by $V_h$. The space $V_h$ is not infinite-dimensional but still very large. Solving a discretized PDE in this space is typically very expensive! In reduced order modeling we utilize the fact that slightly different choices of parameters $\mu$ will give qualitatively similar solutions. We can therefore perform a few simulations in the full space $V_h$ and then make successive simulations cheaper by learning from the past simulations. A crucial concept in this is the solution manifold.

Solution manifold

To any PPDE and a certain parameter set $\mathbb{P}$ we associate a solution manifold:

\[\mathcal{M} = \{z(\mu):F(z(\mu);\mu)=0, \mu\in\mathbb{P}\}.\]

A motivation for reduced order modeling is that even though the space $V_h$ is of very high-dimension, the solution manifold will typically be a very small space. The image below shows a two-dimension solution manifold^[1] embedded in $V_h\equiv\mathbb{R}^3$:

As an example of this consider the one-dimensional wave equation [33]:

\[\partial_{tt}^2q(t,\xi;\mu) = \mu^2\partial_{\xi\xi}^2q(t,\xi;\mu)\text{ on }I\times\Omega,\]

where $I = (0,1)$ and $\Omega=(-1/2,1/2)$. As initial condition for the first derivative we have $\partial_tq(0,\xi;\mu) = -\mu\partial_\xi{}q_0(\xi;\mu)$ and furthermore $q(t,\xi;\mu)=0$ on the boundary (i.e. $\xi\in\{-1/2,1/2\}$).

The solution manifold is a 1-dimensional submanifold of an infinite-dimensional function space:

\[\mathcal{M} = \{(t, \xi)\mapsto{}q(t,\xi;\mu)=q_0(\xi-\mu{}t;\mu):\mu\in\mathbb{P}\subset\mathbb{R}\}.\]

If we provide an initial condition $u_0$, a parameter instance $\mu$ and a time $t$, then $\xi\mapsto{}q(t,\xi;\mu)$ will be the momentary solution. If we consider the time evolution of $q(t,\xi;\mu)$, then it evolves on a two-dimensional submanifold $\bar{\mathcal{M}} := \{\xi\mapsto{}q(t,\xi;\mu):t\in{}I,\mu\in\mathbb{P}\}$.

In reduced order modeling we try to find an approximation to this solution manifolds. Neural networks offer a way of doing so efficiently!

General workflow

In reduced order modeling we aim to construct an approximation to the solution manifold and that is ideally of a dimension not much greater than that of the solution manifold and the solved so-called reduced equations in the small space. This approximation to the solution manifold is performed in the following steps:

1. Discretize the PDE.
2. Solve the discretized PDE for a certain set of parameter instances $\mu\in\mathbb{P}$.
3. Build a reduced basis with the data obtained from having solved the discretized PDE. This step consists of finding two mappings: the reduction $\mathcal{P}$ and the reconstruction $\mathcal{R}$.

The third step can be done with various machine learning (ML) techniques. Traditionally the most popular of these has been Proper orthogonal decomposition (POD), but in recent years autoencoders have also become a popular alternative [34].

After having obtained $\mathcal{P}$ and $\mathcal{R}$ we still need to solve the reduced system. Solving the reduced system is typically referred to as the online phase in reduced order modeling. This is sketched below:

The online phase is applying the mapping $\mathcal{NN}$ in the low-dimensional space in order to predict the next time step. Crucially this step can be made very cheap when compared to the full-order model.

References