Euclidean Embedding of Reaction Networks for Dimensionality Reduction of Chemical Spaces.

Exploring complex chemical reactions is challenging due to the high dimensionality of potential energy surfaces (PESs). Stoichiometry-preserving reaction networks (RNs), constructed using bond-breaking and bond-forming rules, provide a discrete representation of chemical space with an underlying low-dimensional metric structure. Moreover, these RNs resemble regular lattices with a small degree of random edge rewiring. In this work, we show that these RNs coupled with weighted-path metrics can be embedded into low-dimensional Euclidean spaces, producing a significant and chemically informed dimensionality reduction of the PES. Across CHNO RNs spanning diverse stoichiometries and sizes, we compare linear embedding methods, including classical multidimensional scaling (cMDS) and landmark MDS (LMDS), with nonlinear approaches such as metric MDS (mMDS), Isomap, t-distributed stochastic neighbor embedding (t-SNE), and kernel principal component analysis (KPCA). Incorporating kinetic heuristics as edge weights does not significantly increase embedding error or the optimal embedding dimensionality relative to constant edge weights. Although mMDS provides the best distance preservation, it scales poorly with network size. To address this limitation, we introduce a distance-preserving encoder that combines LMDS initialization with neural network refinement and achieves mMDS-level accuracy at a computational cost comparable to LMDS. The resulting embeddings exhibit high fidelity (Kendall τ > 0.84, normalized stress σ < 0.09, and R2 > 0.95) and reduce the dimensionality of the reactive chemical space by a factor of 3-17 relative to the underlying PES. The optimal embedding dimension scales as O(log⁡n) with the number of nodes n. Randomization experiments indicate that the observed near-Euclidean structure is specific to RNs and is related to their chemically constrained topologies and symmetric edge weights, in contrast to general sparse graphs. Pathway discovery experiments further demonstrate that shortest-path searches in the embedding space recover chemically feasible pathways consistent with those identified in the original RNs. These results establish low-dimensional Euclidean embeddings of RNs as a compact and searchable representation of chemical reaction spaces.