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We aim to constrain the gas density and temperature distributions as well as gas masses in several T Tauri protoplanetary disks located in Taurus. We use the 12CO, 13CO, and C18O (2-1) isotopologue emission observed at 0.9 with the IRAM NOrthern Extended Millimeter Array (NOEMA) as part of the MPG-IRAM Observatory Program PRODIGE (PROtostars and DIsks: Global Evolution PIs: P. Caselli & Th. Henning). Our sample consists of Class II disks with no evidence of strong radial substructures. We use thesedata to constrain the thermal and chemical structure of these disks through theoretical models for gas emission. To fit the combined optically thick and thin CO line data in Fourier space, we developed the DiskCheF code, which includes the parameterized disk physical structure, machine-learning (ML) accelerated chemistry, and the RADMC-3D line radiative transfer module. A key novelty of DiskCheF is the fast and feasible ML-based chemistry trained on the extended grid of the disk physical-chemical models precomputed with the ANDES2 code. This ML approach allows complex chemical kinetics models to be included in a time-consuming disk fitting without the need to run a chemical code. We present a novel approach to incorporate chemistry into disk modeling without the need to explicitly calculate a chemical network every time. Using this new disk modeling tool, we successfully fit the 12CO, 13CO, and C18O (2-1) data from the CI, CY, DL, DM, DN, and IQ Tau disks. The combination of optically thin and optically thick CO lines allows us to simultaneously constrain the disk temperature and mass distribution, and derive the CO-based gas masses. These values are in reasonable agreement with the disk dust masses rescaled by a factor of 100 as well as with other indirect gas measurements.