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Title: A differential equation-based framework for magnetic inversions to address challenges with high susceptibility and remanence. Magnetic data are ubiquitous in mineral exploration. They are often inverted under the assumption that the magnetization is purely induced and in the direction of the geomagnetic field. If remanence or self-demagnetization are present, this assumption can lead to erroneous recovered models. Magnetic vector inversion (MVI) allows for a varying direction of magnetization but triples the number of model parameters and increases the non-uniqueness of the inverse problem. Inversion to account for self-demagnetization requires the same number of parameters as traditional methods but introduces additional ambiguity due to non-linearity in the forward modeling. To address these challenges, we introduce a finite-volume based approach within SimPEG that is capable of handling self-demagnetization effects and remanence simultaneously. We then focus on improving methods for inverting magnetic data to recover subsurface distributions of magnetization and high susceptibility separately. We introduce improvements to magnetic vector inversion in Cartesian coordinates to facilitate the recovery of uniformly magnetized and compact targets. We also show that the developed partial differential equation based formulation drastically improves speed and storage requirements for very large scale problems as compared to commonly used integral methods. We illustrate this by inverting data over the Mt. Isa Inlier region in Australia where we recover a model with 41 million parameters in under two hours. To recover distributions of high susceptibility, we introduce an inversion methodology that utilizes sparse regularization with bound constraints. We also introduce a hybrid-parametric sparse inversion approach for targets with more extreme geometries and very high susceptibilities. We apply the hybrid-parametric method to the Osborne deposit in the southern Mt. Isa Inlier region and show that the results compare favorably with drilling.