Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modeling

TL;DR

DIANO proposes a physics-informed, differentiable autoencoding neural operator that compresses high-dimensional flow fields onto a coarse latent grid and evolves these latent states with a differentiable PDE solver. By decoupling spatial encoding from temporal dynamics and employing Fourier-based functional bases, it achieves mesh-invariant, interpretable latent representations while enabling end-to-end training. The approach is demonstrated across four modeling scenarios and three benchmark problems, showing that latent dynamics reflect underlying physics and that varying PDE fidelities balance accuracy, efficiency, and interpretability. This framework advances reduced-order modeling by embedding physical priors directly into latent space and offering a scalable pathway for physics-guided operator learning and latent PDE discovery.

Abstract

Scientific machine learning has enabled the extraction of physical insights from high-dimensional spatiotemporal flow data using linear and nonlinear dimensionality reduction techniques. Despite these advances, achieving interpretability within the latent space remains a challenge. To address this, we propose the DIfferentiable Autoencoding Neural Operator (DIANO), a deterministic autoencoding neural operator framework that constructs physically interpretable latent spaces for both dimensional and geometric reduction, with the provision to enforce differential governing equations directly within the latent space. Built upon neural operators, DIANO compresses high-dimensional input functions into a low-dimensional latent space via spatial coarsening through an encoding neural operator and subsequently reconstructs the original inputs using a decoding neural operator through spatial refinement. We assess DIANO's latent space interpretability and performance in dimensionality reduction against baseline models, including the Convolutional Neural Operator and standard autoencoders. Furthermore, a fully differentiable partial differential equation (PDE) solver is developed and integrated within the latent space, enabling the temporal advancement of both high- and low-fidelity PDEs, thereby embedding physical priors into the latent dynamics. We further investigate various PDE formulations, including the 2D unsteady advection-diffusion and the 3D Pressure-Poisson equation, to examine their influence on shaping the latent flow representations. Benchmark problems considered include flow past a 2D cylinder, flow through a 2D symmetric stenosed artery, and a 3D patient-specific coronary artery. These case studies demonstrate DIANO's capability to solve PDEs within a latent space that facilitates both dimensional and geometrical reduction while allowing latent interpretability.

Figures (8)

• Figure 1: (a) Schematic of the proposed DIANO framework for spatiotemporal modeling. (b) Fourier Neural Operator (FNO), a neural operator using Fourier basis functions for spatial mapping. (c) Nonlinear dimensionality reduction (Static Mapping) with DIANO.
• Figure 2: DIANO architectural variants for each modeling scenario. The encoder compresses inputs via spatial downsampling after each Fourier layer, and the decoder reconstructs outputs via upsampling. Downsampling and upsampling are implemented with AvgPool2D and ConvTranspose2D, respectively. (a) Nonlinear Dimensionality Reduction with Temporal Marching: An unsteady PDE is solved in latent space from $t^n$ to $t^{n+1}$. (b) Geometrical Reduction with Temporal Marching: A 2D input is compressed to 1D, evolved via a given 1D PDE, and decoded back to 2D. (c) Many-to-One Functional Mapping: Three velocity components (u, v, and w) are independently encoded and the 3D PPE equation is solved in latent space to produce the latent pressure field, which is then decoded to full resolution.
• Figure 3: Benchmark flow problems considered in this study. (a) Flow past a 2D circular cylinder. (b) Flow through an idealized, symmetric 2D arterial stenosis. (c) Flow through a patient-specific stenosed left anterior descending (LAD) coronary artery. (d) Transient inlet flow rate ($cm^3/s$) is employed for both cases (b) and (c). The region enclosed by the dotted box indicates the domain used for training and evaluating variants of the autoencoding architectures.
• Figure 4: Nonlinear dimensionality reduction - Static Mapping. Comparison of vorticity latent space structure across different autoencoding methodologies, namely CNO, CNN-AE, NN-AE, and DIANO. The vorticity field of the von Kármán vortex street behind a cylinder is used as the benchmark case. (a) Ground-truth vorticity field from the test dataset, illustrating asymmetric vortex shedding. (b) DIANO: effect of varying the number of Fourier modes within the Fourier layer on the latent space structure with a compression ratio of 4. (c) DIANO: influence of different compression ratios (CR) on the latent space (with 8 Fourier modes). (d) CNO: latent representations at four compression levels. (e) CNN-AE: latent space structure at the compression ratio of 4 with nearly constant value of $0.36$ (failed) using the CNO color legend. (f) NN-AE: temporal evolution of four randomly selected bottleneck modes.
• Figure 5: Nonlinear Dimensionality Reduction with Temporal marching. Comparison of vorticity latent space structure of the DIANO framework (compression ratio 16, 16 Fourier modes) across different VTE simplified formulations solved within the latent space. The vorticity field of the von Kármán vortex street behind a cylinder is used as the benchmark case. (a) Ground truth data from the test dataset, where the input and output flow fields correspond to time instants $t^{n}$ and $t^{n+1}$, respectively. (b) Latent-space vortical structures at $t^{n}$ and $t^{n+1}$ computed by the PDE solver, and (c) the corresponding predicted output flow field at $t^{n+1}$ for 2D linearized VTE. Latent-space vortical structures at $t^{n}$ obtained using: (d) 2D Stokes flow (linearized VTE without the convection term), (e) 2D inviscid linearized VTE, (f) 1D linearized VTE along the streamwise direction $x$, and (g) 1D linearized VTE along the normal direction $y$.