Adaptive tensor train metadynamics for high-dimensional free energy exploration

Abstract

A key challenge for molecular dynamics simulations is efficient exploration of free energy landscapes over relevant collective variables (CV). Common methods for enhancing sampling become prohibitively inefficient beyond only a few CVs; in the case of the widely-used metadynamics method, the computational cost of evaluating and storing the bias potential grows exponentially with the number of dimensions. Here, we introduce TT-Metadynamics, in which the accumulated sum of Gaussian functions in the original metadynamics method is periodically compressed into a low-rank tensor train (TT) representation. The TT enables efficient memory use and prevents the computational cost of evaluating the bias potential from increasing with simulation time. We present a "sketching" algorithm that allows us to construct the TT with linear scaling in the number of CVs. Applied to benchmark systems with up to 14 CVs, the accuracy of TT-Metadynamics matches or exceeds that of standard metadynamics in long simulations, particularly in systems with high barriers. These results establish TT-Metadynamics as a scalable and effective method for computing free energies that are functions of several CVs.

Figures (15)

• Figure 1: Tensor diagrams illustrating tensor trains (TT). (a) TT representation of the $D$-dimensional tensor defined in Eq. \ref{['eq:tt']}. Each vertical solid leg represents a (discrete) coefficient index $i_k$, and each horizontal solid leg represents an (discrete) auxiliary index $\alpha_k$. A horizontal connection between two cores indicates contraction over the associated auxiliary index. (b) Functional TT representation for the bias potential, as in Eq. \ref{['functional_TT']}, showing the contraction of the coefficient tensor with a product of univariate basis functions $\phi_{i_k}^{(k)}(x_k)$ for each collective variable $x_k$. Each vertical dashed leg represents a (continuous) coordinate $x_k$, and correlations between different dimensions are mediated by the horizontal solid auxiliary indices.
• Figure 2: Tensor diagram illustrating the closed form \ref{['coeff_tns_closed_form2']} of $\mathcal{P}_{\text{bias}}^{\text{MetaD}}$. Here we omit the scalar weights $h_i$ in the tensor diagram for simplicity.
• Figure 3: Workflow of the TT-Metadynamics algorithm, illustrating the steps from sampling to bias potential construction.
• Figure 4: 2D structures of alanine dipeptide (a), trialanine (b), ditryptophan (c), and AIB$_9$ (d) with annotated dihedral angles.
• Figure 5: Evolution of free energy of alanine dipeptide during a 50 ns TT-Metadynamics simulation. 2D free energies are approximated as the negative of the instantaneous bias potential, and the examples shown immediately follow sketching (performed every 1 ns). Top left: 1 ns; top right: 2 ns; bottom left: 10 ns; bottom right: 50 ns.