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

2026-08-13

AI Summary

Accurately identifying fractional-order differential equations from noisy data is highly challenging, as high-frequency noise is amplified by fractional differentiation and the derivative order is unknown. This work proposes a weak-form-based discovery framework that mitigates noise by transferring the fractional operator onto smooth test functions through an adjoint-consistent weak formulation. The method jointly searches for the equation structure over both discrete term types and continuous order spaces, efficiently estimating coefficients via a combination of Pareto-front subset selection, branch-aware differential evolution, and ridge regression. Evaluated on multiple benchmark problems, the approach recovers concise and accurate equations from both clean and noisy data, demonstrating significantly improved robustness and reduced computational cost compared to strong-form methods and neural network baselines.

Abstract

Fractional partial differential equations describe nonlocal dynamics, but discovering them from noisy data is difficult because fractional differentiation amplifies high-frequency measurement noise and the derivative orders are unknown. We propose Weak-Pareto, which combines an adjoint-consistent weak formulation of fractional terms with Pareto-based subset selection over discrete term types and continuous fractional orders. For linear right-hand-side terms, the adjoint transfers fractional operators from measured fields to smooth test functions, replacing noise-sensitive pointwise differentiation with smoothing integration; for nonlinear terms, the noise-suppression effect is partial yet useful. Coefficients are fitted by ridge regression within a branch-aware differential-evolution search over the orders. The support size is then selected at the validation-error-complexity elbow. We show that the variance of fixed linear right-hand-side weak features vanishes under grid refinement, whereas noise amplification in pointwise fractional features increases with derivative order. Across fractional advection-diffusion, reaction-diffusion, and Burgers benchmarks, Weak-Pareto recovers parsimonious structures from clean and noisy measurements. In controlled advection-diffusion and Burgers comparisons, it retains the correct support at every tested multiplicative-noise level, whereas the unregularised strong-form counterpart largely fails once noise is introduced; this advantage persists under additive Gaussian noise. Ablations show that the weak library drives noise robustness and that continuous-order Pareto search avoids the support-selection failure of a dense fixed dictionary. On the advection-diffusion benchmark, Weak-Pareto yields more consistent operator recovery and substantially lower measured runtime than a contemporary neural baseline.

Problem

• fractional differential equations
• noisy data
• noise amplification
• unknown derivative orders
• data-driven discovery

Innovation

• fractional differential equations
• weak formulation
• Pareto-based subset selection
• noise robustness
• data-driven discovery