User:Xemizt/mtd

High-dimensional approach
Typical (single-replica) MTD simulations can include up to 3 CVs, even using the multi-replica approach, it is hard to exceed 8 CVs, in practice. This limitation comes from the bias potential, constructed by adding Gaussian functions (kernels). It is a special case of the kernel density estimator (KDE). The number of required kernels, for a constant KDE accuracy, increases exponentially with the number of dimensions. So MTD simulation length has to increase exponentially with the number of CVs to maintain the same accuracy of the bias potential. Also, the bias potential, for fast evaluation, is typically approximated with a regular grid. The required memory to store the grid increases exponentially with the number of dimensions (CVs) too.

A high-dimensional generalization of metadynamics is NN2B. It is based on two machine learning algorithms: the nearest-neighbor density estimator (NNDE) and the artificial neural network (ANN). NNDE replaces KDE to estimate the updates of bias potential from short biased simulations, while ANN is used to approximate the resulting bias potential. ANN is a memory-efficient representation of high-dimensional functions, where derivatives (biasing forces) are effectively computed with the backpropagation algorithm.

An alternative method, exploiting ANN for the adaptive bias potential approximation, uses mean potential forces for the estimation. This methods are also a high-dimensional generalization of the adaptive bias force (ABF) method. Additionally, the training of ANN is improved using the Bayesian regularization, and the error of approximation can be inferred by training an ensemble of ANNs.

Free energy estimator
The finite size of the kernel makes the bias potential to fluctuate around a mean value. A converged free energy can be obtained by averaging the bias potential. The averaging is started from $$t_\text{diff}$$, when the motion along the collective variable becomes diffusive:

=
$$\bar F(\vec s) = - \frac{1}{t_\text{sim} - t_\text{diff}} \int^{t_\text{sim}}_{t_\text{diff}} \!\!\!\!\!V_\text{bias}(\vec s, t)\, dt + C$$ =====

Well-tempered metadynamics
Well-tempered metadynamics (WT-MTD) is a modification of the original metadynamics algorithm, where the scale of the Gaussian kernel is varied during the simulations.

Test $$

$$F(\vec s\,) = -\frac{T + \Delta T}{\Delta T} \!\lim_{t_\text{sim} \to \infty}\!\!\! V_\text{bias}(\vec s\,) + C $$

Well-tempered ensemble metadynamics
Well-tempered ensemble metadynamics (WTE-MTD)

Transition-tempered metadynamics
Transition-tempered metadynamics (TT-MTD)

Adaptive Gaussian metadynamics
Adaptive Gaussian metadynamics (AG-MTD)

Mutiple-walker metadynamics
Multiple-walker metadyanmics (MW-MTD)

Parallel tempering metadynamics
Parallel tempering metadynamics (PT-MTD)

$$\begin{align} P_{i \leftrightarrow j} &= \min(1, \exp(\Delta \epsilon_{i \leftrightarrow j})) \\ \Delta \epsilon_{i \leftrightarrow j} &= (\beta_j - \beta_i)(V(\vec q_i) - V(\vec q_j)) \\ &+ \beta_i\, (V_{\text{bias},i}(\vec s_i) - V_{\text{bias},i}(\vec s_j)) \\ &+ \beta_j (V_{\text{bias},j}(\vec s_j) - V_{\text{bias},j}(\vec s_i)) \\ \end{align}$$

Bias-exchange metadynamics
Bias-exchange metadynamics (BE-MTD)

$$\begin{align} \Delta \epsilon_{i \leftrightarrow j} = \beta\, (V_{\text{bias},i}(\vec s_i) &- V_{\text{bias},i}(\vec s_j)\\       +\, V_{\text{bias},j}(\vec s_j) &- V_{\text{bias},j}(\vec s_i)) \end{align}$$

Collective-variable tempering metadynamics
Collective-varialbe tempering metadynamics (CVT-MTD)

Parallel bias metadynamics
Parallel bias metadynamics (PB-MTD)

Replica state exchange metadynamics
Replica state exchange metadynamics (RSE-MTD)

Reconnaissance metadynamics
Reconnaissance metadynamics (RC-MTD)

Flux-tempered metadynamics
Flux-temepered metadynamics (FT-MTD)

Replica-averaged metadynamics
Replica-average metadynamics (RA-MTD)

Ensemble-biased metadynamics
Ensemble-biased metadynamics (EB-MTD)

Path integral metadynamics
Path integral metadynamics (PI-MTD)

Discreet metadynamics
Discreet metadynamics (D-MTD)

Lagrangian metadynamcis
Lagrangian metadynamics (L-MTD)