LaSIPDE: Latent-Space Identification of Partial Differential Equations from indirect, high-dimensional measurements

Discovering governing equations from data is a central challenge in scientific machine learning, particularly when observations are high-dimensional and the underlying state variables are not directly accessible. In this work, we introduce a framework for data-driven discovery of partial differential equations (PDEs) from indirect high-dimensional observations. The proposed approach combines nonlinear representation learning through an autoencoder with sparse identification of governing equations in the latent space, enabling simultaneous model reduction and PDE discovery while preserving spatial structure. Unlike existing methods that either operate on observable variables or discover latent ordinary differential equations, our framework identifies PDEs directly in the learned latent coordinates. We validate the approach on high-dimensional observations generated from Burgers and Korteweg-de Vries (KdV) systems, where the true state variables are intentionally hidden. In both cases, the method successfully recovers the correct dynamical operators, including diffusion, nonlinear advection, and dispersive terms. Although the recovered coefficients differ due to latent coordinate transformations, we show both theoretically and empirically that the discovered equations are dynamically equivalent to the ground-truth systems up to an affine transformation. These results demonstrate that governing PDEs can be recovered from indirect, high-dimensional data without access to the physical state variables, providing a foundation for interpretable model discovery in realistic measurement settings.