SCOPE: Observation-Conditioned Full-Target Prediction for Sparse PDE 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.
Related
- Uncertainty-Aware Coordinated Multi-Ramp Metering Based on an Action-Conditioned Traffic Prediction Operator Under Sparse Observations
- Direct large-area observation of subsurface plastic activity in conditioned copper electrodes
- Coincidence-conditioned Compton-modulation recovery from annihilation Bell pairs
- Optical-parameter-conditioned one-shot autofocus for multi-objective-lens microscopy
- Macroscopic freezing of a vector Gaussian free field conditioned to avoid a ball
BibTeX
@misc{xu2026scope,
title = {{SCOPE: Observation-Conditioned Full-Target Prediction for Sparse PDE Inference}},
author = {Ruichen Xu and Siyao Wang and Fang Wan and Jiacheng Qiu and Wenhan Gao and Jiaxing Zhang and Linsey Pang and Ravid Shwartz-Ziv and Prakhar Mehrotra and Yann LeCun and Yuefan Deng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.36527},
eprint = {2609.36527},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36527},
}
RIS
TY - UNPB
TI - SCOPE: Observation-Conditioned Full-Target Prediction for Sparse PDE Inference
AU - Ruichen Xu
AU - Siyao Wang
AU - Fang Wan
AU - Jiacheng Qiu
AU - Wenhan Gao
AU - Jiaxing Zhang
AU - Linsey Pang
AU - Ravid Shwartz-Ziv
AU - Prakhar Mehrotra
AU - Yann LeCun
AU - Yuefan Deng
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.36527
UR - https://arxiv.org/abs/2609.36527
AB - 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.
ER -