Weak-Pareto Discovers Fractional PDEs Robustly from Noisy Data.

Discovering fractional partial differential equations (PDEs) from noisy experimental data is a significant challenge, primarily because fractional differentiation inherently amplifies high-frequency measurement noise, and the exact derivative orders are often unknown. Existing methods struggle with this noise sensitivity. Researchers have developed Weak-Pareto, a robust data-driven approach that addresses these issues. It integrates an adjoint-consistent weak formulation for fractional terms with a Pareto-based subset selection strategy. For linear terms, the adjoint transfers fractional operators to smooth test functions, replacing noise-sensitive pointwise differentiation with noise-suppressing integration. While less complete for nonlinear terms, this still provides substantial benefits. The method fits coefficients using ridge regression within a differential-evolution search for orders, then selects the optimal support size. Experiments across various fractional benchmarks (advection-diffusion, reaction-diffusion, Burgers) show Weak-Pareto consistently recovers parsimonious structures even with significant noise, outperforming unregularized strong-form methods and neural baselines in robustness and operator recovery.