A fast discovery framework of fractional transport equations through fractional-order operator linearization

Data-driven discovery of fractional differential equations can recover compact governing laws for anomalous transport and memory-dependent dynamics, but fractional-order estimation remains computationally expensive because the orders enter the operators nonlinearly. We develop a fast sparse-regression framework that locally linearizes fractional operators with respect to their orders. Each fractional order is decomposed into a reference value and a first-order correction; the corresponding linearization column is appended to an augmented regression system, and the orders are updated iteratively within a bounded radius motivated by the first-order truncation error. For a space–time fractional advection–dispersion benchmark, the method recovers the correct equation structure under 0%–25% noise in 4–7 s, whereas a differential-evolution baseline using the same surrogate, candidate library, and STRidge configuration requires 690–1601 s and yields larger parameter errors. Coupling the framework with the generative candidate library EqGPT is also practical, requiring about 65 s for 80 sampled structures. Applied to MODFLOW–MODPATH particle-tracking data in a heterogeneous Gamma-distributed conductivity field, the method identifies an effective time-fractional advection law consistent with the purely advective transport mechanism. Applied to MADE field tracer data, it recovers a spatial fractional advection–dispersion–reaction equation that captures the main plume migration, spreading and attenuation. These results show that fractional-order operator linearization can significantly reduce the cost of fractional equation discovery while remaining compatible with both prescribed and generated candidate libraries and applicable to simulated and field transport data.