On the generalization ability of coarse-grained molecular dynamics models for non-equilibrium processes

Abstract

One essential goal of constructing coarse-grained molecular dynamics (CGMD) models is to accurately predict non-equilibrium processes beyond the atomistic scale. While a CG model can be constructed by projecting the full dynamics onto a set of resolved variables, the dynamics of the CG variables can recover the full dynamics only when the conditional distribution of the unresolved variables is close to the one associated with the particular projection operator. In particular, the model's applicability to various non-equilibrium processes is generally unwarranted due to the inconsistency in the conditional distribution. Here, we present a data-driven approach for constructing CGMD models that retain certain generalization ability for non-equilibrium processes. Unlike the conventional CG models based on pre-selected CG variables (e.g., the center of mass), the present CG model seeks a set of auxiliary CG variables based on the time-lagged independent component analysis to minimize the entropy contribution of the unresolved variables. This ensures the distribution of the unresolved variables under a broad range of non-equilibrium conditions approaches the one under equilibrium. Numerical results of a polymer melt system demonstrate the significance of this broadly-overlooked metric for the model's generalization ability, and the effectiveness of the present CG model for predicting the complex viscoelastic responses under various non-equilibrium flows.

Figures (9)

• Figure 1: Diagram illustrating the neural network architecture for the CG conservative force and the memory kernel. (a) The construction of CG potential function $U$ and memory kernel $\mathbf K$. Initially, $\mathbf Q$ is converted into a local environment matrix $\{\tilde{\mathbf Q}_\mu \}_{\mu=1}^{Mm}$. Sub-networks, illustrated in (b), map $\tilde{ \mathbf Q}_\mu$ to a local feature $\mathbf D_\mu$ and generalized coordinate $\hat{\mathbf Q}_\mu$. Finally, the totel potential is constructed by Eq. \ref{['eq:total-potential']}, i.e. $U= \sum_\mu^{Mm} \tilde{U}( D_\mu)$. The total memory kernel $\mathbf K$ is constructed with the state-dependent component of the memory kernel derived from $\hat{\mathbf Q}_\mu$ using Eq. \ref{['eq:Xi_markovian']} and the time-dependent component $\bm \Lambda$. (b) The sub-networks map $\tilde{ \mathbf Q}_\mu$ to a local feature $D_\mu$ and generalized coordinate $\hat{\mathbf Q}_\mu$. The neighbour of the $\mu-$th CG coordinate is denoted by $\mathcal{N}_\mu= \{\alpha,\cdots,\nu,\cdots,\beta\}$. (b1) The $k-$th row of generalized local environment matrix embeds the relative information between $\mu-$th coordinate and its $k-$th neighbor (labeled as the $\nu-$th coordinate), including type embedding $a_\nu$, $a_\mu, b_{\nu\mu}$, and distance information. (b2) The $K_1\times K_2$ symmetry perserving feature $\mathbf D_\mu$ is constructed by $(\mathbf G_{1,\mu})^T$, $\hat{\mathbf Q}_\mu$, $\hat{\mathbf Q}_\mu^ T$ and $\mathbf G_{2,\mu}$. (b3) The generalized coordinate is constructed from the local environment embedding matrix $\mathbf G_{3,\mu}$ and relative position $\bar{\mathbf Q}_\mu$, which is the last three columns of $\tilde{\mathbf Q}_\mu$.
• Figure 2: The Frobenius norm of the second-moment difference $\left \Vert \mathbf C(y) - \mathbf C(y')\right \Vert_F$ of the various CG models under the steady reverse Poiseuille flow generated by external force $f_0=0.01$ (upper) and $f_0=0.005$ (lower), where $y$ and $y'$ are represented by the horizontal and vertical axis, respectively. (a) and (d) CGCOM; (b) and (e) CG3; (c) and (f) CG4.
• Figure 3: The pairwise conservative force difference $\left \vert F_{ij}^\text{neq}(Q) - F_{ij}^{\text{eq}}(Q,y)\right \vert$ for various CG models, which loosely quantifies the generalization ability of the CG free energy $U(\mathbf Q)$. (a) CGCOM; (b) CG3 with $(i,j) = (1,1)$; (c-f) CG4 with $(i,j) = (1,1), (2,2), \cdots, (4,4)$.
• Figure 4: The difference of the variation of the fluctuation force $\vert K^{ij}_\text{neq}(Q)- K^{ij}_{\text{eq}}(Q,y)\vert$ for different CG models, which loosely quantifies the generalization ability of the CG memory term $\mathbf K(\mathbf Q, t)$ at $t=0$. The sub-figure labels are the same as Fig. \ref{['fig:cf']}.
• Figure 5: The development of the reverse Poiseuille flow under the external force $f_0 = 0.01$ predicted by the full MD and three CG models.

Theorems & Definitions (3)

• Definition B.1
• Proposition B.2
• proof