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This research introduces SCOPE, a novel neural operator framework that deterministically recovers complete physical fields from sparse observations using full-field latent prediction coupled with physical reconstruction. By decoupling optimal latent prediction from field reconstruction through quadratic risk decomposition, the method outperforms both diffusion-based solvers and existing mask-aware operators across diverse partial differential equation settings. The approach offers a significant advance in computational physics by enabling accurate, single-pass inference without iterative sampling or retraining backbones.
SCOPE: Observation-Conditioned Full-Target Prediction for Sparse PDE Inference
arXiv:2609.36527v2 Announce Type: replace 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.
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.
34 pages, including supplementary material. Code: this https URL. Author affiliation updated
Machine Learning (cs.LG); Computational Physics (physics.comp-ph)
arXiv:2609.36527 [cs.LG]
(or arXiv:2609.36527v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.36527 Focus to learn more arXiv-issued DOI via DataCite
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