SCOPE: Observation-Conditioned Full-Target Prediction for Sparse PDE Inference

2026-09-28

AI Summary

This study addresses the non-uniqueness challenge in physical field recovery from sparse observations by proposing an observability-aware latent variable embedding framework. The method couples full-field latent prediction with physical reconstruction, utilizing a shared decoder to guide representation learning for efficient, deterministic single-step inference. Furthermore, through quadratic risk decomposition theory, it elucidates the suboptimality bounds between latent prediction and field reconstruction, along with the transferability conditions for decoder refinement. Evaluated across ten tasks spanning five PDE scenarios, the proposed approach significantly outperforms mask-aware neural operators, achieves lower errors than diffusion models, and supports decoder adaptation without retraining the backbone network.

Abstract

Recovering complete physical fields from sparse observations is challenging because the measurements may not uniquely determine the underlying state. Diffusion-based PDE solvers address this problem through iterative sampling whereas neural operators provide deterministic one-pass predictions. We propose SCOPE (Sparse-Context Observability-aware Predictive Embeddings) to recover complete PDE fields from sparse observations by coupling full-field latent prediction with physical reconstruction. A shared decoder reconstructs fields from both predicted and complete-view representations so that representation learning is guided by both physical recovery and latent matching. We derive a quadratic risk decomposition at fixed teacher-decoder pairs showing why optimal latent prediction need not yield optimal field reconstruction. We also establish sufficient conditions for decoder improvements on complete inputs to transfer to recovery from partial observations. Experiments across five PDE settings show that SCOPE outperforms mask-aware neural operators on all ten forward and inverse tasks and achieves lower errors than those reported for diffusion-based solvers including DiffusionPDE and FunDPS. Decoder-only adaptation further improves recovery without retraining the backbone while retaining deterministic single-pass inference.

Problem

• sparse PDE inference
• physical field recovery
• sparse observations
• full-target prediction

Innovation

• Sparse PDE Inference
• Latent Prediction
• Shared Decoder
• Risk Decomposition
• Neural Operators