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

arXiv:2506.20181v2 Announce Type: replace Abstract: 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 gove