OTGPD

What OTGPD does

The Optimal Transport GP Dimer (OTGPD) is a saddle point search method that combines the GP-Dimer approach with an adaptive oracle-calling strategy. The key question it answers: when should the optimizer trust the GP and when should it call the oracle?

In the standard GP-Dimer, the oracle is called at every outer iteration regardless of how confident the GP is. OTGPD adds a decision threshold: if the GP-predicted force is small enough relative to the observed forces so far, trust the GP and skip the oracle call.

Adaptive threshold

The oracle-calling threshold adapts as the search progresses:

\[T_{\mathrm{GP}} = \max\!\left(\frac{\min_{k} \lVert \mathbf{F}^{\mathrm{true}}_{k} \rVert}{d},\; \frac{T_{\mathrm{dimer}}}{10}\right)\]

where \(\min_{k} \lVert \mathbf{F}^{\mathrm{true}}_{k} \rVert\) is the smallest true force observed so far, \(d\) is the divisor controlling how aggressive the GP trust is (default 5.0), \(T_{\mathrm{dimer}}\) is the dimer convergence tolerance, and the floor \(T_{\mathrm{dimer}} / 10\) ensures the GP is never trusted with tolerance tighter than 10% of the convergence criterion.

Why this formula works: early in the search, forces are large (far from the saddle), so min(F_true) / divisor is large, and the oracle is called frequently. As the search converges, forces decrease, T_gp tightens, and the GP must predict very small forces before being trusted. This naturally transitions from oracle-intensive exploration to GP-dominated refinement.

Why the floor T:sub:`dimer`/10: without a floor, as forces approach zero, T_gp would also approach zero, and the GP would never be trusted (defeating the purpose). The floor ensures the GP is always given a chance to skip oracle calls when predicted forces are within 10% of the convergence tolerance.

HOD training data management

OTGPD uses History-Ordered Data (HOD): training points are kept in temporal order, and when the dataset exceeds fps_history, the oldest points are pruned via FPS.

Why temporal ordering: in saddle point search, the optimizer follows a single trajectory from the starting point toward the saddle. Older points are geometrically distant from the current position and contribute less to local GP accuracy. Pruning old points keeps the training set focused on the current region.

This differs from spatial FPS (used in minimization and standard GP-Dimer), which selects points spread across the entire training set regardless of when they were collected.

Initial rotation

OTGPD inherits the dimer’s rotation+translation structure. When a good initial orientation is available (from NEB climbing image tangent or Hessian eigenmode), rotation can be skipped entirely (max_rot_iter = 0).

This is common in practice: one first runs NEB to find the MEP, then refines the highest-energy image with OTGPD to get a precise transition state. The NEB path tangent at the climbing image provides the dimer orientation.

Relationship to GP-Dimer

OTGPD reduces to the standard GP-Dimer when:

• fps_history is large enough that HOD pruning never activates

• The adaptive threshold saturates at T_dimer / 10

On the LEPS surface, both methods converge in ~13 oracle calls starting from 0.05 A displaced from the known saddle along the negative eigenmode. The adaptive threshold provides robustness: if the starting point is far from the saddle (where the GP is initially poor), OTGPD calls the oracle more frequently, while GP-Dimer would waste calls on poor GP predictions.

When to use OTGPD vs GP-Dimer

• Starting near the saddle (e.g., NEB refinement): both methods perform similarly. Use GP-Dimer for simplicity.

• Starting far from the saddle (exploratory search): OTGPD’s adaptive threshold avoids wasting oracle calls on poor early GP predictions.

• Long trajectories (many iterations): OTGPD’s HOD pruning keeps the training set focused, avoiding the Cholesky cost growth that affects GP-Dimer.

Configuration Reference

Parameter	Default	Description
conv_tol	0.1	Force norm convergence threshold
dimer_sep	0.005	Half-length of the dimer
max_iter	200	Maximum outer iterations
max_oracle_calls	0 (unlimited)	Oracle call budget
rff_features	0 (exact GP)	RFF feature count for PredModel
fps_history	0 (use all)	FPS subset size (HOD management)
t_gp_divisor	5.0	Divisor for adaptive threshold
trust_radius	0.1	Trust region radius