The Degeneracy Distillery

Models are usually parameterized by scientific convention, not according to the combinations of parameters that the data can actually distinguish. When observations depend on only a few combinations of many parameters, simulation-based inference is unnecessarily high-dimensional: neural estimators spend expensive simulator calls learning redundant directions. Even at modest dimensionality, nonlinear degeneracies can make inference highly simulation-intensive and obscure the quantities actually being measured. We present the degeneracy distillery, which automatically discovers these combinations and expresses them in closed form, simultaneously reducing the effective parameter dimension and the simulation budget required for downstream inference. From parameter–simulation pairs alone, without likelihood evaluations, simulator gradients, or an observed data realization, the method learns an information-aware coordinate map across the prior and distills it into compact symbolic expressions. A joint objective determines the number of identifiable coordinates while driving the transformed information metric toward the identity. The resulting coordinates are both low-dimensional and interpretable: they reveal how the original parameters combine to control the data and restrict downstream inference to the directions the data actually constrain. Across scientific applications, the method recovers known structure from simulations alone: $R_0\propto \beta /\gamma$ from noisy epidemic trajectories and chirp mass from gravitational-wave inspirals. For full inspiral–merger–ringdown waveforms, it instead discovers total mass and an asymmetric mass combination that yields better-conditioned and better-calibrated inference than conventional chirp-mass coordinates. In controlled scaling experiments, the method recovers rank-1 and rank-2 structure embedded in parameter spaces with up to 32 dimensions. Neural posterior estimation in the discovered coordinates tracks an oracle given the true parameterization, while inference in the original parameters degrades with ambient dimension. By removing redundant parameter directions before density estimation, the distilled coordinates substantially reduce the simulation budget required for simulation-based inference: $9$–$14\times$ fewer simulations at matched validation log-probability and approximately $2\times$ fewer at matched CRPS.