MCO-PDE: AI discovers hidden physics equations from sparse multi-source data

The paper introduces MCO-PDE, a method that addresses the challenge of discovering governing PDEs when only sparse or fragmented observational data is available, a common scenario in physics and engineering. The framework first trains independent neural networks (surrogates) on each data source to capture local dynamics. Then, a soft-competitive weighting scheme dynamically assesses the credibility of each dataset and aggregates a consensus global coefficient. This is coupled with a genetic algorithm that searches over possible functional forms of the PDE, simultaneously identifying both structure and parameters. The approach is designed to handle datasets that share underlying physics but differ in initial or boundary conditions, making it practical for real-world multi-experiment scenarios.

Results across seven test cases demonstrate that MCO-PDE can accurately recover canonical equations—such as Burgers', Korteweg–de Vries, and Navier–Stokes—using as few as 50 observations per dataset, even in the presence of irregular boundaries, heterogeneous coefficients, and two- or three-dimensional domains. The framework was further validated on real-world data from wave-tank experiments, extracting physically meaningful laws. This work represents a significant step toward automating scientific discovery by fusing heterogeneous data sources, potentially enabling researchers to infer governing equations from scattered experimental setups without requiring a single comprehensive dataset.