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DeePCG: constructing coarse-grained models via deep neural networks
Linfeng Zhang, Jiequn Han, Han Wang, Roberto Car, Weinan E
TL;DR
Coarse-grained models need accurate free-energy representations for extensive variables without restricting interactions to low body order. DeePCG uses a symmetry-preserving neural network trained from atomistic data to construct a many-body CG potential; for liquid water, it reproduces two-, three-, and higher-order oxygen correlations while accelerating sampling.
Problem
Accurate free-energy functions for extensive coarse-grained variables are complex and nonlinear, and existing machine-learning work has mainly represented atomistic potential-energy surfaces.
Method
DeePCG uses a symmetry-preserving neural-network representation of the many-body CG free-energy potential, trained with a force-matching formulation from atomistic data.
Results
The liquid-water model reproduces two-, three-, and higher-order oxygen correlation functions very well, with a 7.5-times acceleration over DeePMD.
Takeaways & Limitations
DeePCG provides an accurate framework for parameterizing CG potentials and sampling CG configurations via molecular dynamics without ad hoc interaction-order approximations.
Takeaways & Limitations
The implementation has force discontinuities from the sharp cutoff, fixed angular-neighbor count, and abrupt changes in sorted atomic lists.
Abstract
from arXiv · showhide
We introduce a general framework for constructing coarse-grained potential models without ad hoc approximations such as limiting the potential to two- and/or three-body contributions. The scheme, called Deep Coarse-Grained Potential (abbreviated DeePCG), exploits a carefully crafted neural network to construct a many-body coarse-grained potential. The network is trained with full atomistic data in a way that preserves the natural symmetries of the system. The resulting model is very accurate and can be used to sample the configurations of the coarse-grained variables in a much faster way than with the original atomistic model. As an application we consider liquid water and use the oxygen coordinates as the coarse-grained variables, starting from a full atomistic simulation of this system at the ab-initio molecular dynamics level. We found that the two-body, three-body and higher order oxygen correlation functions produced by the coarse-grained and full atomistic models agree very well with each other, illustrating the effectiveness of the DeePCG model on a rather challenging task.
I. INTRODUCTION
Coarse-graining serves either low-dimensional free-energy analysis or extensive-variable configurational sampling. DeePCG addresses the latter by learning a many-body CG potential that preserves system symmetries without ad hoc interaction-order restrictions.
- I. INTRODUCTION: Molecular dynamics coarse-graining targets either Landau free-energy evaluation with few variables or faster sampling with extensive coarse-grained variables.The first uses scalar or low-dimensional vector variables; the second reduces atomistic coordinates while retaining system-size-dependent dimensionality.
- I. INTRODUCTION: Extensive-variable coarse-grained free energies are complex and nonlinear, so accurate representations often require substantial physical or chemical intuition.Machine learning may automate this representation, but prior approaches largely focused on atomistic potential-energy surfaces rather than CG free-energy surfaces.
- I. INTRODUCTION: DeePCG generalizes Deep Potential and DeePMD to represent the free-energy surface in coarse-grained-variable space as a many-body CG potential.The framework requires no ad hoc approximations beyond the neural-network model itself.
- I. INTRODUCTION: For liquid water, the method uses oxygen-site point particles as coarse-grained variables and compares their behavior with an underlying ab-initio molecular-dynamics model.The application tests whether a single particle can represent each water molecule in a challenging many-body setting.
A. Basic Theory
The CG potential is defined to reproduce the atomistic model’s configurational distribution after projection onto reduced variables. Constructing it requires choosing a representation and optimizing its parameters, while dynamical recovery remains outside this work’s scope.
- A. Basic Theory: Coarse-grained variables are reduced coordinates ξ(q) with M < dN, representing either finite-dimensional order parameters or extensive replacements for molecular objects.Their configurational distribution is obtained by projecting the atomistic configurational distribution onto CG-variable space.
- A. Basic Theory: A good CG potential should accurately reproduce the full configurational distribution of the CG variables, although validation commonly uses only two- and three-body correlations.The full distribution is difficult to test directly.
- A. Basic Theory: Constructing a CG model involves selecting an appropriate CG-potential representation and optimizing the parameters that define it.These two choices distinguish alternative coarse-graining schemes.
- A. Basic Theory: Even an exact CG potential does not provide a closed deterministic equation of motion because dN − M degrees of freedom are missing.Recovering CG dynamical information generally requires additional assumptions such as time-scale separation.
B. CG potential representation
DeePCG represents the CG potential with a symmetry-preserving neural network built from local particle environments. A finite cutoff makes the trained model’s computational cost scale linearly with system size.
- B. CG potential representation: The CG potential is represented by a neural network U_w(ξ) that takes only generalized coordinates as input and is optimized without human intervention.The representation is designed to preserve the symmetry properties of the target CG potential.
- B. CG potential representation: For point particles with positional dependence, the coarse-grained variables are the coordinates of the CG particles.The described implementation is limited to CG objects behaving as point particles.
- B. CG potential representation: Each particle’s potential contribution is computed from descriptors of neighboring particles within a cutoff radius, followed by a fully connected feedforward neural network.The descriptors are produced by a local-frame transformation before entering the network.
- B. CG potential representation: Translational and rotational symmetries are preserved by the local-frame transformation, while permutation symmetry follows from sorted descriptors and shared subnetworks.The construction therefore encodes the relevant geometric and particle-exchange symmetries in the representation.
- B. CG potential representation: The DeePCG simulation cost scales linearly with system size because the model uses a finite cutoff and sums local contributions.The additive local structure enables direct application to larger systems after training.
C. CG potential optimization
DeePCG optimizes a neural-network representation of the coarse-grained potential by force matching. Its instantaneous-force loss avoids expensive constrained simulations while retaining the same minimizer as the mean-force loss.
- Neural-network representation: The CG potential is represented by local-environment contributions computed from descriptors and fully connected neural networks.A local-frame transformation encodes neighboring CG particles before each sub-network predicts a potential contribution.
- Force matching: Force matching trains the model by minimizing the discrepancy between predicted CG forces and mean forces estimated from atomistic data.The mean force is defined through conditional equilibrium averages on hypersurfaces of fixed CG variables.
- Instantaneous-force loss: Instantaneous-force training replaces constrained or restrained ensemble averages with forces sampled from equilibrium atomistic configurations.This approach is computationally easier and relies on representative configurations sampled under the appropriate thermodynamic conditions.
- Instantaneous-force loss: The instantaneous and mean-force loss functions have the same minimizer because their difference is independent of the model parameters.The instantaneous force decomposes into the mean force plus a zero-average random error.
- Optimization: The framework supports nonlinear CG mappings, while stochastic gradient descent efficiently optimizes the highly non-convex neural-network loss.The authors report that different local minima found by SGD approximate the target physics similarly well.
III. COARSE-GRAINING OF LIQUID WATER
The authors apply DeePCG to liquid water by replacing oxygen-centered atomistic environments with effective CG particles. The trained model is evaluated through CG simulations against atomistic structural distributions.
- Application setup: The water application coarse-grains an ab initio DFT-based liquid-water simulation into effective particles centered on oxygen atoms.The motivation is to represent the intrinsically many-body DFT potential with a coarse-grained model without relying on manually designed interaction forms.
- Application setup: The underlying AIMD dataset uses the PBE0 functional, self-consistent Tkatchenko-Scheffler dispersion, periodic boundaries, and deuterons for a 0.5 fs timestep.These details specify the atomistic reference simulation used for training and comparison.
- Evaluation: DeePMD-sampled oxygen configurations are reported to be in almost perfect agreement with AIMD across O-O RDFs, O-O-O ADFs, and averaged local Steinhardt-parameter distributions.This supports using DeePMD-generated configurations for oxygen-coordinate training as essentially indistinguishable from AIMD-generated data.
- Model construction: DeePCG uses a 6 Å cutoff, full radial-angular descriptors for the 16 nearest oxygen CG particles, and radial descriptors for the remaining neighbors.Each local environment defines a sub-network with four hidden layers whose widths decrease across layers.
- Training: The model is trained by minimizing the instantaneous force-matching loss, using atomistic oxygen forces and Adam-based stochastic gradient descent.The reported implementation uses batch size 4 and an exponentially decaying learning rate.
- Evaluation: CG NVT simulations use an AIMD snapshot as initialization, the AIMD volume and temperature, analytical CG forces, and a Langevin thermostat.The damping time is τ = 0.1 ps, and an additional simulation uses 512 CG variables.
IV. DISCUSSION
DeePCG reproduces two-, three-, and higher-order oxygen structural statistics of atomistic water models while scaling to larger systems and accelerating simulations. Its consistency improves with longer, more diverse training trajectories, although force discontinuities remain an implementation limitation.
- Structural agreement: DeePCG reproduces oxygen two-, three-, and higher-order correlation functions of the atomistic DeePMD and underlying AIMD models.The comparisons include O-O RDFs, O-O-O ADFs, and Steinhardt order-parameter distributions.
- Model consistency: 99.3% of model deviations are below 50 meV/Å when training uses six independent 2.5 ns trajectories.The deviation does not become more significant at larger CG-force magnitudes, and independently initialized models generate indistinguishable configurational distributions.
- Model consistency: Shorter training trajectories increase model deviations, confirming that longer trajectories better approximate the ensemble average used in training.The comparison covers total training lengths of 2 ns, 5 ns, and 15 ns.
- Computational cost and scalability: 7.5 times: the current DeePCG implementation accelerates DeePMD while retaining linear cost scaling with system size.The model was also tested in a 512-bead system eight times larger than the training system, with only slightly weaker RDF structure.
- Implementation limitation: Force discontinuities arise from the sharp cutoff, fixed angular-neighbor limit, and abrupt neighbor-list sorting changes.Training reduces these discontinuities below thermal fluctuations, allowing them to be subsumed by the stochastic thermostat during canonical sampling, but the potential remains piece-wise continuous.
V. CONCLUSION AND FUTURE WORK
DeePCG is presented as a general tool for parameterizing coarse-grained potentials and sampling coarse-grained configurations via molecular dynamics. The authors identify broader applications and future tests beyond liquid water.
- DeePCG parameterizes coarse-grained potentials and samples coarse-grained configurations via molecular dynamics.
- The procedure’s generality is expected to make DeePCG useful for a wide variety of tasks.
- For water, the authors propose investigating whether the introduced coarse-grained model can describe ice as well as liquid water.
Appendix A: Definition of the local averaged Steinhardt parameters
The local averaged Steinhardt parameters characterize bond orientational order using neighboring particles and spherical harmonics. Their expansion includes higher-order angular correlations, including four-body terms.
- The local Steinhardt parameter q_l(i) describes the bond orientational order of particle i in a condensed environment.
- Its definition uses neighboring particles, spherical harmonics Y_lm(r̂_ij), and a distance-dependent switching function s(r_ij).
- The modified parameter uses r_min = 0.31 nm and r_max = 0.36 nm and is more sensitive for distinguishing crystal structures.
- The full expansion contains four-body terms, so the parameter distribution incorporates four-body angular correlations.