Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability

We study operator relevance in data-driven partial differential equation (PDE) discovery.

Sparse residual methods can select terms that improve residual fit, but residual contribution is not the same as functional necessity.

We formalize this distinction through counterfactual operator interventions, where a candidate term is deleted or perturbed and the factual and intervened trajectories, or observables, are compared.

The resulting theory gives six reusable results.

A residual--counterfactual gap theorem shows that deletion effects are governed by the inverse linearized PDE map, not by residual magnitude alone.

A certified decision theorem gives error margins for relevance, irrelevance, and abstention under neural or numerical surrogate error.

An aliasing theorem characterizes experiment-dependent non-identifiability through the null space of the operator-evaluation design.

A constraint-manifold theorem shows that operators vanishing on invariant constraint classes cannot be identified from trajectories restricted to those classes.

A pruning-consistency theorem proves that sparse screening followed by counterfactual deletion recovers the functionally relevant support under a recall and margin condition.

An observable-level adjoint theorem extends relevance testing from full-state deviations to scientific quantities of interest.

Validation experiments test these mechanisms on synthetic PDEs with known support and on public geophysical fields from atmospheric reanalysis and NOAA OISST.

The real-data results are reported as operator-surrogate diagnostics, not as unconditional recovery of physical laws.

The framework provides a rigorous diagnostic layer for distinguishing residual usefulness from counterfactual operator relevance within a specified library, experiment class, norm, and tolerance.