Latent Dynamics
Definition
Compress a high-dimensional field into a low-dimensional latent code with an encoder, learn the temporal evolution in latent space (ODE / neural ODE / RNN), then decode back to physical space. The dominant architecture for nonlinear reduced-order modeling.
What it is
Three steps: encoder: u -> z (low-dimensional), a dynamics model z(t) -> z(t+dt), and a decoder: z -> u. Because evolution happens in a latent space of a few dozen dimensions, long rollouts are extremely cheap, and the nonlinear encoder can represent curved solution manifolds that POD cannot.
Key points
- Why it beats POD: for traveling waves and moving shocks, where the linear subspace dimension is large, a nonlinear encoder represents the same physics with far fewer DOFs.
- Choosing the dynamics model: linear (DMD/Koopman) is interpretable but weak; neural ODEs are flexible but expensive to train; RNN/LSTM are simple but drift over long horizons.
- Interpretability cost: latent variables usually carry no physical meaning, so conservation laws cannot be imposed directly and must enter as physics regularization in the loss.
- Training recipe: train the autoencoder for reconstruction first, then freeze it and train the dynamics — usually more stable than end-to-end.
Common pitfalls / boundaries
- Smaller latent dimension is not always better: too small loses physics and shows up as "low reconstruction error but diverging rollout".
- Decoding error is amplified by the dynamics, so always evaluate rollout error, not reconstruction error.