ADAPTIVE=DIFF: Initial Undesirable Gaussian

[zbaibg]

Dear PLUMED developers,

When using ADAPTIVE=DIFF, the SIGMA parameter should be set to a time over which the Gaussian width is averaged from the fluctuations of the CVs. Typically, this averaging time needs to be larger than the autocorrelation time of the CV to ensure good statistics. In my case, this corresponds to approximately ~8 ps.

However, the PACE parameter can be chosen to be much smaller in order to place Gaussians more frequently—for example, 0.5 ps in my simulations—otherwise the metadynamics would converge too slowly. Under these conditions, the width of the very first Gaussian is determined from insufficient statistics and is usually much smaller than the true fluctuation. In well-tempered metadynamics, this leads to the deposition of a narrow, abnormally high Gaussian. Subsequent Gaussians, being much wider and lower, cannot efficiently compensate for this artifact. As a result, the region where the first Gaussian was placed suffers from serious statistical imbalance. This issue is particularly problematic in multi-walker simulations.

I would like to suggest allowing the averaging procedure for the Gaussian width to begin before the time specified by UPDATE_FROM. This would ensure that the first Gaussian benefits from a sufficient averaging period and thus has a more reliable width.

I plan to attempt an implementation of this idea myself, but I would greatly appreciate any guidance, advice, or feedback from the developers.

Thanks!

[zbaibg]

I have implemented a version that introduces the new metadynamics option “PRECAL_DIFF” and developed corresponding test scripts based on the v2.10.0 branch. At present, the implementation is limited to METAD and does not extend to PBMETAD or other variants. If this functionality is considered useful, I would be happy to help with merging it into the master branch.

https://github.com/zbaibg/plumed2/tree/v2.10.0_bg

The following report summarizes the test results.

PRECAL_DIFF Test Results (OU Trajectory)

Setup

• Source: run_precal_suite.sh
• Trajectory: Ornstein-Uhlenbeck process generated by generate_trajectory.py
  • Parameters: dt = 0.005 ps, τ = 0.5 ps, σ = 0.15 nm
• MetaD Configuration:
  • ADAPTIVE=DIFF, PRECAL_DIFF toggled ON/OFF
  • PACE=5, TEMP=300, BIASFACTOR=5, SIGMA=4000 (20 ps)
• UPDATE_FROM Values: 0, 0.25, 0.5, 1, 2, 4, 5, 10, 20, 40 ps
• Reproduction: ./run_precal_suite.sh | tee test_output

Results: First Hill Sigma Comparison

UPDATE_FROM (ps)	OFF_sigma	ON_sigma
0.00	0.00106309	0.00106309
0.25	0.00116244	0.00536729
0.50	0.00093972	0.00859008
1.00	0.00099863	0.02147966
2.00	0.00056206	0.02876307
4.00	0.00088638	0.05148014
5.00	0.00054712	0.05620057
10.00	0.00039714	0.11195997
20.00	0.00280476	0.13397390
40.00	0.00139940	0.15776509

Key Observations

1. Validation at UPDATE_FROM=0: Identical behavior with/without PRECAL_DIFF confirms correct implementation
2. Systematic Enhancement: PRECAL_DIFF consistently produces larger first-hill sigma values, addressing early-time underestimation in ADAPTIVE=DIFF
3. Scaling with UPDATE_FROM: As UPDATE_FROM increases, ON_sigma approaches values more representative of the true OU process width (σ = 0.15 nm)
4. Physical Interpretation: Pre-averaging from t=0 effectively mitigates small-sigma artifacts that occur when hills are deposited before sufficient sampling

Implementation Files

• run_precal_suite.sh: Main test script that generates inputs, executes OFF/ON comparisons, and produces summary statistics
• test_output: Console output containing the tabulated results
• generate_trajectory.py: OU trajectory generator for consistent test conditions