Reconstructing Multi-Scale Physical Fields from Extremely Sparse Measurements with an Autoencoder-Diffusion Cascade

TL;DR

The paper tackles the challenge of reconstructing full multi-scale physical fields from extremely sparse measurements. It introduces Cascaded Sensing (Cas-Sensing), a hierarchical framework that first uses a neural operator–based functional autoencoder to recover the large-scale structure, then employs a conditional diffusion model to add fine-scale details conditioned on the recovered structure and sparse data, with measurement fidelity enforced by a manifold-constrained gradient. This two-stage cascade reduces ill-posedness and improves robustness and generalization across sensor configurations, demonstrated on cylinder flow, sea-surface waves, and global SST data. The results show accurate, coherent reconstructions from minute observation ratios and highlight Cas-Sensing’s potential for practical data-driven sensing in science and engineering.

Abstract

Reconstructing full fields from extremely sparse and random measurements is a longstanding ill-posed inverse problem. A powerful framework for addressing such challenges is hierarchical probabilistic modeling, where uncertainty is represented by intermediate variables and resolved through marginalization during inference. Inspired by this principle, we propose Cascaded Sensing (Cas-Sensing), a hierarchical reconstruction framework that integrates an autoencoder-diffusion cascade. First, a neural operator-based functional autoencoder reconstructs the dominant structures of the original field - including large-scale components and geometric boundaries - from arbitrary sparse inputs, serving as an intermediate variable. Then, a conditional diffusion model, trained with a mask-cascade strategy, generates fine-scale details conditioned on these large-scale structures. To further enhance fidelity, measurement consistency is enforced via the manifold constrained gradient based on Bayesian posterior sampling during the generation process. This cascaded pipeline substantially alleviates ill-posedness, delivering accurate and robust reconstructions. Experiments on both simulation and real-world datasets demonstrate that Cas-Sensing generalizes well across varying sensor configurations and geometric boundaries, making it a promising tool for practical deployment in scientific and engineering applications.

Figures (10)

• Figure 1: Overview of the proposed Cascaded Sensing (Cas-Sensing) based on an autoencoder-diffusion cascade.
• Figure 2: An example of 2D cylinder flow velocity fields. The dataset contains 96 different boundary configurations of circular cylinders, each associated with 100 temporal snapshots.
• Figure 3: t-SNE visualization of latent representations learned by the functional autoencoder for cylinder flow fileds.
• Figure 4: The reconstructed principal structures and kernel density estimates for RMSE on the reference sample across 100 randomly chosen meshes.
• Figure 5: The full-field reconstruction results of cylinder flow fields.