Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modeling

Scientific machine learning has enabled the extraction of physical insights and data-driven modeling of high-dimensional spatiotemporal data, yet achieving physically interpretable latent representations and computationally efficient surrogates remains an open challenge. We propose the DIfferentiable Autoencoding Neural Operator - DIANO, an autoencoding neural operator framework that constructs visualizable coarse-grid latent spaces for both dimensionality and geometric reduction across varying spatial discretizations, with governing equations enforced directly within the latent space. Built upon neural operators, DIANO achieves this through an encoding neural operator that spatially coarsens the high-dimensional input functions into the latent representation, and a decoding neural operator that reconstructs the original inputs via spatial refinement. We assess DIANO's latent representation and performance against baselines, including the Convolutional Neural Operator and standard autoencoders. Furthermore, a fully differentiable partial differential equation (PDE) solver is integrated as the sole input-output functional mapping operator within the latent space, enabling end-to-end training with governing physics prescribed a priori through parametric PDEs. Various PDE formulations are investigated, including the 2D unsteady advection-diffusion and the 3D Pressure--Poisson equation, revealing that the fidelity of the embedded PDE relative to the true physics governs the learned latent representation and reconstruction accuracy. Benchmark problems include flow past a 2D cylinder, flow through a 2D symmetric stenosed artery, and a 3D patient-specific coronary artery, showing accurate reconstruction of high-fidelity spatio-temporal fields through low-fidelity latent PDE evolution at reduced computational cost, while yielding coherent, spatially organized, and meaningful latent structures.