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What is LaSDI?
What is LaSDI?
LaSDI is a framework for implementing reduced order models. As discussed in the paper (above), many current reduced-order model techniques fall into the LaSDI framework, such as DMD, PCA, and others.
As computer simulations of various systems (physical, biological, etc.) become more complex, the time to run them increases. For instance, simple fluid simulations can take hours, if not days, to achieve high resolution results. While these results are accurate, the computational time makes them intractable in various applications, such as engineering, inverse problems, etc.
Reduced-order modeling seeks to speed up the computational time while minimizing the incurred error. Various techniques have been proposed (see paper) which achieve relative error on the order of 0.0001% or smaller. However, they only decrease simulation time by a factor of 2-10.
LaSDI approaches the problem from a different perspective and achieves speed up times on the order of 100-1000, while incurring a relative error around 1%. Here we can see the clear trade-off between speed and error. Thus, in applications where error is less of an issue, LaSDI can be used to obtain faster, high-resolution simulations.
vid.mp4A brief explanation of LaSDI. LLNL-PRES-826500
How does LaSDI work?
How does LaSDI work?
The problem:
The problem:
Consider a complex simulation in which we want to understand how the initial conditions impact the full simulation. LaSDI offers a method to quickly generate multiple high-resolution simulations in minimal amounts of time.
For example, how does a shock wave depend on the height of the initial wave?
General Overview:
General Overview:
LaSDI works by "learning" a compression/decompression of the simulation. So, for a simulation with 10,000 spatial points, LaSDI will learn how to represent these as 3-5 points (latent variables). Just as the 10,000 points are governed by physical laws, which can be represented as equations, we expect the latent variables to also have equations to represent their evolution in time. So, using computational techniques to approximate these equations, we can use the resulting equations to generate new simulations in a fraction of the time. We no longer require a simulation of all 10,000 spatial points, but rather only 3-5 latent variables which can then be decompressed to generate a high-resolution simulation.
4 primary steps:
4 primary steps:
Generating Full-Order Model (FOM) training data
Training a Data-Compression technique to extract a latent-space
Learning the dynamics of the reduced order space
Generating ROMs for new parameter values.
We explain these in detail below.
LaSDI currently works with both POD (LaSDI-LS) and Neural Network (LaSDI-NM) compression techniques, allowing for versatility in both diffusion and advection dominated problems.
- Data Generation
For LaSDI to work, we need full simulation data. This can be selected in various ways. In the paper above, we use a uniform grid on our parameter space. However, greedy-algorithms can be used to minimize the number of training values needed to optimize performance.
- Training the Compressor/Decompressor
We use the full-order data, along with a neural-network or other technique, to create a bottleneck as shown above (2). The orange component represents the compression technique which reduces our full-order simulation to 3-5 variables. Similarly, the yellow component represents the decompression technique, which restores the full image. We are most interested in the resulting bottleneck, in which we only have 3-5 variables representing the full simulation.
- Dynamics Identification
We use regression, similar to SINDy, to approximate the dynamics that govern the latent variables. The resulting equations might be representative of physical laws; however, this is not necessarily the case. Still, the equations will represent how the latent variable (and thus the full simulation) evolves over time.
- Prediction
Because we are interested in how the initial condition impacts the simulation, we first need to find how the initial condition translates to the latent variables. So, we take our initial configuration, pass it through the compression technique, and then use the approximated equations to evolve the variables in time. Once we have simulated for the prescribed amount of time, we can decompressed the latent variables to generate a new full simulation with minimally incurred error.
Examples
(scroll to bottom for results)Examples
LaSDI-LS on 2D Burgers (not accurate due to POD technique)
LaSDI-NM on 2D Burgers (accurate due to Autoencoder technique)
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