Gradient Navigation on an Invariant Manifold for Crystal Structure Prediction

Crystal structure prediction (CSP) is often bottlenecked by stochastic moves that revisit the same funnels and spend many costly relaxations escaping local traps. We introduce Fingerprint‑Surrogate Flow (FSF), a differentiable global search that descends a learned potential on an invariant fingerprint manifold and projects that motion back to atoms via the analytic Jacobian. Each structure is mapped to a rotation/translation/permutation‑invariant fingerprint f(r); an on‑the‑fly Gaussian‑process surrogate provides a smooth objective combining predicted energy, uncertainty, soft attraction to an anchor set of diverse low‑energy minima, and repulsion from visited basins. Short micro‑trajectories follow the projected gradient, yielding monotone descent in Φ (Lyapunov property) with fully analytic, symmetry‑aware “forces.” The anchor term leverages derivatives of an assignment‑invariant fingerprint distance, creating barrier‑agnostic bridges that promote funnel‑to‑funnel moves. We will present the Jacobian‑projection derivation, ablations, scaling to multi‑walker runs, and an open‑source implementation. FSF offers a principled, end‑to‑end differentiable alternative that turns landscape knowledge into forces for fast, reliable CSP.