  We introduce the Binless Multidimensional Thermodynamic Integration (BMTI)
method for nonparametric, robust, and data-efficient density estimation. BMTI
estimates the logarithm of the density by initially computing log-density
differences between neighbouring data points. Subsequently, such differences
are integrated, weighted by their associated uncertainties, using a
maximum-likelihood formulation. This procedure can be seen as an extension to a
multidimensional setting of the thermodynamic integration, a technique
developed in statistical physics. The method leverages the manifold hypothesis,
estimating quantities within the intrinsic data manifold without defining an
explicit coordinate map. It does not rely on any binning or space partitioning,
but rather on the construction of a neighbourhood graph based on an adaptive
bandwidth selection procedure. BMTI mitigates the limitations commonly
associated with traditional nonparametric density estimators, effectively
reconstructing smooth profiles even in high-dimensional embedding spaces. The
method is tested on a variety of complex synthetic high-dimensional datasets,
where it is shown to outperform traditional estimators, and is benchmarked on
realistic datasets from the chemical physics literature.
