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Model Reduction and Time Series
An inherent challenge for many practical PDE solvers is the large dimensionality of the problem. Our model \mathcal{P} is typically discretized with \mathcal{O}(n^3) samples for a 3 dimensional problem (with n denoting the number of samples along one axis), and for time-dependent phenomena we additionally have a discretization along time. The latter typically scales in accordance to the spatial dimensions, giving an overall number of samples on the order of \mathcal{O}(n^4). Not surprisingly, the workload in these situations quickly explodes for larger n (and for practical high-fidelity applications we want n to be as large as possible).
One popular way to reduce the complexity is to map a spatial state of our system \mathbf{s_t} \in \mathbb{R}^{n^3} into a much lower dimensional state \mathbf{c_t} \in \mathbb{R}^{m}, with m \ll n^3. Within this latent space, we estimate the evolution of our system by inferring a new state \mathbf{c_{t+1}}, which we then decode to obtain \mathbf{s_{t+1}}. In order for this to work, it’s crucial that we can choose m large enough that it captures all important structures in our solution manifold, and that the time prediction of \mathbf{c_{t+1}} can be computed efficiently, such that we obtain a gain in performance despite the additional encoding and decoding steps. In practice, due to the explosion in terms of unknowns for regular simulations (the \mathcal{O}(n^3) above) coupled a super-linear complexity for computing a new state, working with the latent space points \mathbf{c} quickly pays off for small m.
However, it’s crucial that encoder and decoder do a good job at
reducing the dimensionality of the problem. This is a very good task for
DL approaches. Furthermore, we then need a time evolution of the latent
space states \mathbf{c}, and for most
practical model equations, we cannot find closed form solutions to
evolve \mathbf{c}. Hence, this likewise
poses a very good problem for learning methods. To summarize, we’re
facing to challenges: learning a good spatial encoding and decoding,
together with learning an accurate time evolution. Below, we will
describe an approach to solve this problem following Wiewel et al.
{cite}wiewel2019lss &
{cite}wiewel2020lsssubdiv, which in turn employs the
encoder/decoder of Kim et al. {cite}bkim2019deep.
| ```{figure} resources/timeseries-lsp-overview.jpg |
|---|
| height: 200px |
| name: timeseries-lsp-overview |
For time series predictions with ROMs, we encode the state of our system with an encoder f_e, predict the time evolution with f_t, and then decode the full spatial information with a decoder f_d. ```
Reduced Order Models
Reducing the order of computational models, often called reduced order modeling (ROM) or model reduction, as a classic topic in the computational field. Traditional techniques often employ techniques such as principal component analysis to arrive at a basis for a chosen space of solution. However, being linear by construction, these approaches have inherent limitations when representing complex, non-linear solution manifolds. And in practice, all “interesting” solutions are highly non-linear.
\text{arg min}_{\theta} | f_d( f_e(x;\theta_e) ;\theta_d) - x |_2^2
f_e: \mathbb{R}^{n^3} \rightarrow \mathbb{R}^{m}
f_d: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n^3}
separable model
Time Series
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