Robust data-driven discovery of fractional differential equations via weak formulations and Pareto-based subset selection

What the math gives to ML

The paper's transferable asset is an adjoint weak formulation for fractional operators: instead of differentiating noisy measurements, it moves the fractional derivative onto smooth test functions and evaluates an integral against the measured field. This creates noise-averaging features whose variance decreases with grid refinement, unlike strong-form fractional derivatives whose noise amplification grows with order. A practical neural transfer is to replace pointwise fractional PDE residuals in a neural dynamics model or PINN with several precomputed weak residual projections, optionally combined with sparse ridge selection over operator types and continuous fractional orders. The resulting loss should be substantially more stable when observations are noisy or temporally undersampled.