Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations

2026-05-13

AI Summary

This work addresses the challenge of efficiently and accurately solving both forward and inverse problems of partial differential equations (PDEs) under extremely sparse observational conditions—down to 3% data availability—where conventional methods struggle. The authors propose a unified neural framework that integrates variational autoencoders, latent diffusion models, and contrastive learning within a compressed latent space, augmented by a novel PDE-aware variance-preserving denoising algorithm. This approach significantly enhances both inference accuracy and computational efficiency. For the first time, it enables high-fidelity bidirectional PDE solving and zero-shot super-resolution prediction, achieving state-of-the-art performance across multiple benchmarks while substantially reducing computational costs and supporting high-resolution modeling over continuous spatiotemporal domains.

Abstract

Partial differential equations (PDEs) are fundamental for modeling complex natural and physical phenomena. In many real-world applications, however, observational data are extremely sparse, which severely limits the applicability of both classical numerical solvers and existing neural approaches. While neural methods have shown promising results under moderately sparse observations, their inference efficiency at high resolutions is limited, and their accuracy degrades substantially in the extremely sparse regime. In this work, we propose the Di-BiLPS, a unified neural framework that effectively handle both forward and inverse PDE problems under extremely sparse observations. Di-BiLPS combines a variational autoencoder to compress high-dimensional inputs into a compact latent space, a latent diffusion module to model uncertainty, and contrastive learning to align representations. Operating entirely in this latent space, the framework achieves efficient inference while retaining flexible input-output mapping. In addition, we introduce a PDE-informed denoising algorithm based on a variance-preserving diffusion process, which further improves inference efficiency. Extensive experiments on multiple PDE benchmarks demonstrate that Di-BiLPS consistently achieves SOTA performance under extremely sparse inputs (as low as 3%), while substantially reducing computational cost. Moreover, Di-BiLPS enables zero-shot super-resolution, as it allows predictions over continuous spatial-temporal domains.

Problem

• Partial Differential Equations
• Sparse Observations
• Neural Solvers
• Inverse Problems
• Forward Problems

Innovation

• latent-space PDE solver
• diffusion-based denoising
• extremely sparse observations
• contrastive representation learning
• zero-shot super-resolution