Autoencoder Reduced-Order Models
An autoencoder learns a nonlinear low-dimensional coordinate system for a simulation, often far more compact than a linear basis.
Nonlinear compression
Linear reduction by proper orthogonal decomposition is limited to flat subspaces. Many systems live on a curved, low-dimensional manifold that a linear basis captures poorly, needing many modes. An autoencoder learns a nonlinear encoder that maps the high-dimensional state to a few latent coordinates, and a decoder that reconstructs the state, capturing the curved manifold directly.
Why fewer coordinates
Because the encoder and decoder are nonlinear, they can represent structures that would require many linear modes. A traveling wave, for instance, needs a large linear basis but only one or two nonlinear latent coordinates. This sharper compression makes the downstream dynamics model smaller and cheaper while remaining accurate.
Modeling latent dynamics
With the state compressed, the evolution of the latent coordinates is modeled separately, by a neural ordinary differential equation, a recurrent network, or sparse regression on the latent variables. Training the encoder and the dynamics together, so the latent space is shaped to have simple dynamics, generally works better than training them in sequence.
Design considerations
- Latent dimension: too small loses fidelity, too large defeats the purpose
- Physical constraints can be imposed on the latent dynamics
- Convolutional encoders suit fields on structured grids
- Reconstruction and dynamics losses must be balanced
Trade-offs
Autoencoder reduced-order models can be dramatically more compact than linear ones, but they are harder to train, less interpretable, and offer weaker guarantees. Where a linear method suffices, it is often preferred for its transparency. Where the manifold is strongly curved, as in many transport-dominated flows, the nonlinear autoencoder earns its added complexity.