  Descriptors are physically-inspired schemes for representing atomistic
systems that play a central role in the construction of models of potential
energy surfaces. Although physical intuition can be flexibly encoded into
descriptor schemes, they are generally ultimately guided only by the spatial or
topological arrangement of atoms in the system. Here, we propose a novel
approach for the optimization of descriptors based on encoding information
about geodesic distances along potential energy manifolds into the
hyperparameters of commonly used descriptor schemes. To accomplish this, we
combine two ideas: (1) a differential-geometric approach for the fast
estimation of approximate geodesic distances; and (2) an information-theoretic
evaluation metric - information imbalance - for measuring the shared
information between two distance measures. Using the MD22 datasets of ethanol,
malonaldehyde, and aspirin, we first show that Euclidean (in Cartesian
coordinates) and geodesic distances are inequivalent distance measures,
indicating the need for updated ground-truth distance measures that go beyond
the Euclidean distance. We then utilize a Bayesian optimization framework to
show that descriptors (in this case, atom-centered symmetry functions) can be
optimized to maximally express a certain type of distance information, such as
Euclidean or geodesic information. We also show that modifying the Bayesian
optimization algorithm to minimize a combined Euclidean+geodesic objective
function can yield descriptors that not only express both Euclidean and
geodesic distance information simultaneously, but in fact resolve substantial
disagreements between descriptors optimized to encode only one type of distance
measure. We discuss the relevance of our approach to the design of more
physically rich and informative descriptors that can encode useful, alternative
information about molecular systems.
