Bi-HYCO: Bi-Objective Cooperative Learning for PDE Parameter Identification under Fragmented Observations

We introduce HyCOP, a modular framework that learns parametric PDE solution operators by composing simple modules (advection, diffusion, learned closures, boundary handling) in a query-conditioned way. Rather than learning a monolithic map, HyCOP learns a policy over short programs - which module to apply and for how long - conditioned on regime features and state statistics. Modules may be numerical sub-solvers or learned components, enabling hybrid surrogates evaluated at arbitrary query times without autoregressive rollout. Across diverse PDE benchmarks, HyCOP produces interpretable programs, delivers order-of-magnitude OOD improvements over monolithic neural operators, and supports modular transfer through dictionary updates (e.g., boundary swaps, residual enrichment). Our theory characterizes expressivity and gives an error decomposition that separates composition error from module error and doubles as a process-level diagnostic.

Discovering PDEs in heterogeneous media requires jointly identifying the governing operator and the unknown spatial fields that parameterize it. These tasks are coupled: changing field placement changes the differential law, while a sufficiently flexible field can conceal structural error on a single trajectory. We present Hypothesize, Evaluate, Refine for PDE Discovery (HER-PDE), a scientific-agent framework that discovers compositional PDE structure together with nonparametric, time-invariant coefficient fields. The Agent analyzes two noisy trajectories generated by different excitations, proposes complete expression-tree hypotheses, and combines creative structural exploration with local candidate refinement. Its Hypothesis Evaluation Interface (HEI) estimates only the fields explicitly declared in each hypothesis, never adds missing terms, and scores structures by bidirectional cross-excitation transfer. The selected law is subsequently audited on a sealed temporal interval. Across five controlled two-dimensional systems observed with 5 percent relative Gaussian state noise, the Agent recovers the generating operator in all five cases, including equivalent signed-field and product-rule parameterizations. Across nine unknown coefficient fields, the recovered fields attain a median Pearson correlation of approximately 0.85 and a median relative L2 error of approximately 0.28. These results show that agent-guided hypothesis refinement can recover heterogeneous governing laws without prescribing a parametric form for their spatial coefficients.

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.