MINT: Shared PDE Identification with Neural Fields and Genetic Programming

Identifying a partial differential equation (PDE) from sparse measurements is difficult because different equations can fit the same observed solution. We propose MINT (Multi-trajectory Identification with Neural fields and Trees), which fits independent neural fields and uses genetic programming (GP) to construct a single equation shared across multiple trajectories. MINT searches for terms and coefficients simultaneously and retains the selected structure while refining the fields and coefficients. We also develop MINT-M, which uses additional family-specific motifs to guide initialization. Across six benchmarks with 30 runs each, MINT recovers the terms of 2D and 3D advection in 86.7% and 43.3% of noise-free runs, respectively, whereas two GP-based baselines recover neither. In the same setting, MINT-M recovers Burgers and Fisher–KPP terms in 56.7% and 63.3% of runs, respectively, compared with no recoveries for MINT. Both methods also recover 2D advection more often than the GP baselines at 1% and 5% noise. With 250 observations per trajectory, both improve field accuracy over the common data-only reconstruction on 2D advection, Burgers, and Fisher at both noise levels. Missing-library and noise tests further show that accurate fields and low residuals alone do not establish recovery of the governing law.