Structured latent dynamics reconstruct fine-scale PDE patterns from coarse data
Latent dynamics for reconstructing fine-scale PDE trajectories from coarse observations
Machine Learning
Summary
Simulations of physical systems often only capture rough snapshots, missing small details and intermediate steps. The authors present Structured Latent Dynamics (SLD), a new method that filling in these missing details by learning how the system evolves from less detailed observations. Their approach combines known mathematical operations with learned corrections to rebuild high-resolution sequences more accurately than other similar methods. This lets them effectively reconstruct detailed behaviors of complex systems like chemical patterns and fluid flows from limited data.
What this means in practice
- •For computational physicists: Recover detailed simulation data from sparse observations to improve analysis of complex physical systems.
- •For weather forecast modelers: Enhance coarse atmospheric model outputs by reconstructing finer scale dynamic details to improve short-term weather predictions.
Authors
Chenhao Si, Zishan Qian, Shing-Tung Yau, Rongling Wu, Ming Yan
Abstract
Coarse PDE observations omit spatial detail and intermediate states of evolving fields. We present Structured Latent Dynamics (SLD), a framework that reconstructs high-resolution trajectories from spatially and temporally downsampled simulations. SLD interpolates the input in time, adds a learned temporal correction and uses the resulting sequence to guide a hidden state on the coarse grid. Each update of the latent state combines a learned gated response with horizontal and vertical central differences and a five-point Laplacian of the state. The three spatial terms have learned channel-wise coefficients. Two Haar wavelet synthesis branches map the hidden state and refined coarse input to the high-resolution field. We evaluate SLD on two-dimensional Gray--Scott, complex Ginzburg--Landau and viscous Burgers systems, reconstructing 100 output frames from 25 input frames with eightfold refinement in each spatial direction. SLD achieves lower mean relative $L_2$ error and mean absolute error than unstructured latent dynamics and ConvLSTM baselines with comparable parameter counts, both on held-out trajectories and on later windows of trajectories used during training. SLD also reduces errors in regions with large reference gradients and across most or all of the evaluated later time intervals. A representative Gray--Scott comparison shows lower spectral error over most intermediate and high wavenumbers. Removing the fixed spatial response increased reconstruction error, supporting its contribution to the SLD architecture.