CARE-PDE: Robust PDE Discovery under Noisy Data–Equation Correspondence

Data-driven partial differential equation (PDE) discovery aims to recover explicit governing equations from spatiotemporal observations. Existing methods are primarily limited to clean data or data corrupted by simple, single-source noise. In practice, however, multiple noise sources can produce heterogeneous contamination, causing some observations to deviate substantially from the underlying dynamics and weakening their correspondence with the governing equation. We formulate this problem as Noisy Data–Equation Correspondence (NDEC) and propose the Correspondence-Aware Robust Equation Discovery for PDEs (CARE-PDE) framework. Specifically, we develop Reliability-Aware Field Reconstruction (RAFR) by combining a novel Cauchy loss with adaptive reliability estimation to reduce the influence of heavily corrupted observations. Building on these estimations, Reliability-Guided Local Legendre Projection (RGLP) further exploits the estimated reliability to construct reliable local supports and recover stable derivatives through orthogonal projection. Local Integral Constraint Aggregation (LICA) is then presented to aggregate the reconstructed strong-form relations into local integral constraints, reducing sensitivity to pointwise estimation errors. Finally, Stability-Selected Sparse Regression (SSR) identifies consistently supported terms to recover sparse governing equations. Experiments on 12 PDE systems across different contamination scenarios show that CARE-PDE recovers the correct equation support in all 36 tasks and achieves lower true-term coefficient errors than the ten state-of-the-art baselines, demonstrating its effectiveness and robustness under the tested NDEC conditions.