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SCOPE bridges sparse observation inference and full-state recovery by coupling latent-space full-field prediction with physical reconstruction using a shared decoder across complete and partial representations. This architecture directly addresses the theoretical misalignment between latent matching loss and true physical recovery error, established via a derived quadratic risk decomposition. Across five PDE benchmarks spanning ten forward and inverse tasks, the model achieves lower error than iterative diffusion solvers like FunDPS while retaining single-pass deterministic inference and modular, decoder-only adaptation.
Single-pass neural operators can actually beat iterative diffusion solvers on sparse PDE recovery, provided latent-space prediction is structurally coupled with physical field decoding.
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.
