Interpretable Test-Time Adaptation of Latent Dynamics from Sparse Sensors

Data-driven models of spatiotemporal dynamics are trained on a limited range of physical parameters and routinely deployed on regimes outside it, where a few point sensors may be the only measurements available. Under such parameter shift, the learned latent dynamics no longer describe the system, and the full-state data needed for retraining the model does not exist. We propose TTA-SINDy-SHRED for sparse identification of nonlinear dynamics with shallow recurrent decoder networks (SINDy-SHRED), enabling efficient test-time adaptation to unseen pa- rameter regimes. The joint training objective shapes the SINDy-SHRED latent representation so that a mechanistic governing equation captures the dynamics across regimes, with all regime variation concentrated in the coefficient matrix of the latent space physics model. At test time, all network weights stay fixed and only two context variables are inferred. Adaptation requires no backprop- agation, keeps the latent dynamics interpretable, and updates the decoder from point measurements alone. We evaluate on shock-forming advection–diffusion, reaction–diffusion pattern formation, and decaying two-dimensional turbulence. In every held-out regime, adaptation from a short calibration window reduces long-horizon forecast error relative to the unadapted model. We further test trans- ferability to both neural-operator and attention-based architectures paired with learned latent dynamics, and TTA-SINDy-SHRED attains the lowest error across forecast horizons.