Causal Operator Discovery in Partial Differential Equations via Counterfactual Physics-Informed Neural Networks

We develop a principled framework for discovering causal structure in partial differential equations (PDEs) using physics-informed neural networks and counterfactual perturbations. Unlike classical residual minimization or sparse regression methods, our approach quantifies operator-level necessity through functional interventions on the governing dynamics. We introduce causal sensitivity indices and structural deviation metrics to assess the influence of candidate differential operators within neural surrogates. Theoretically, we prove exact recovery of the causal operator support under restricted isometry or mutual coherence conditions, with residual bounds guaranteeing identifiability. Empirically, we validate the framework on both synthetic and real-world datasets across climate dynamics, tumor diffusion, and ocean flows. Our method consistently recovers governing operators even under noise, redundancy, and data scarcity, outperforming standard PINNs and DeepONets in structural fidelity. This work positions causal PDE discovery as a tractable and interpretable inference task grounded in structural causal models and variational residual analysis.