Arrow Flow Matching for Reaction Mechanisms

Chemists explain a reaction by drawing arrows. Each arrow moves one pair of valence electrons, and an elementary step made of such arrows creates no atom and no electron. We propose a generative model of these arrows. Its state is an integer bond–electron matrix, and its dynamics is a Markov chain trained by discrete flow matching. The central design choice is the elementary event, the change that one update of the chain applies to the state. The common choice is an edit of one entry of the matrix. We prove that a model which draws every entry independently conserves the electron count exactly only if it is deterministic in every entry, whatever its parameters. Arrow Flow Matching instead takes as its elementary event one move from a four-element alphabet, the matrix form of an arrow that chemists draw. Every move leaves the electron count unchanged, and the chain may stop only where a table of valid per-atom states allows it. Conservation therefore reduces to an algebraic identity and validity to a lookup on one softmax class, and both hold with probability one. As every update is an arrow, the trajectory of the model reads as a mechanism that a chemist can inspect. These guarantees cost almost no accuracy. Arrow Flow Matching is competitive with the most accurate model we train, and at its smallest width it already reaches a higher composite score than every other model at every width.