Latent dynamics for reconstructing fine-scale PDE trajectories from coarse observations

Chenhao Si, Zishan Qian, Shing-Tung Yau, Rongling Wu, Ming Yan

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…