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.

Algorithms

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:

Adaptive kernel algorithms

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.

${\frac {\partial V_{\text{bias}}({\vec {s}}\,)}{\partial t}}=\overbrace {\omega _{0}\exp \!\!\left(\!-{\frac {V_{\text{bias}}({\vec {s}}\,)}{\Delta T}}\right)} ^{\omega ({\vec {s}}\,)}\delta (|{\vec {s}}-{\vec {s}}_{t}|)$ (1)

Test 1

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

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)

Multiple-replica algorithms

Mutiple-walker metadynamics

Multiple-walker metadyanmics (MW-MTD)

Parallel tempering metadynamics

Parallel tempering metadynamics (PT-MTD)

${\begin{aligned}P_{i\leftrightarrow j}&=\min(1,\exp(\Delta \epsilon _{i\leftrightarrow j}))\\\Delta \epsilon _{i\leftrightarrow j}&=(\beta _{j}-\beta _{i})(V({\vec {q}}_{i})-V({\vec {q}}_{j}))\\&+\beta _{i}\,(V_{{\text{bias}},i}({\vec {s}}_{i})-V_{{\text{bias}},i}({\vec {s}}_{j}))\\&+\beta _{j}(V_{{\text{bias}},j}({\vec {s}}_{j})-V_{{\text{bias}},j}({\vec {s}}_{i}))\\\end{aligned}}$

Bias-exchange metadynamics

Bias-exchange metadynamics (BE-MTD)

${\begin{aligned}\Delta \epsilon _{i\leftrightarrow j}=\beta \,(V_{{\text{bias}},i}({\vec {s}}_{i})&-V_{{\text{bias}},i}({\vec {s}}_{j})\\+\,V_{{\text{bias}},j}({\vec {s}}_{j})&-V_{{\text{bias}},j}({\vec {s}}_{i}))\end{aligned}}$

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)