SimPoly: Simulation of Polymers with Machine Learning Force Fields Derived from First Principles

Gregor SimmJean HelieHannes SchulzYicheng ChenGuillem SimeonAnna KuzinaErnesto Martinez-BaezPiero GasparottoGabriele TocciChi Chen

article2025arXiv12 citations

Develops an ab initio machine learning force field that predicts bulk polymer densities and glass transition temperatures more accurately than classical force fields, accompanied by a 130-polymer experimental benchmark.

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Polymers are essential across numerous industries, ranging from aerospace and electronics to medicine and packaging. Designing new polymeric materials computationally could drastically reduce development timelines and costs. However, standard modeling methods face a severe bottleneck: classical force fields rely on tedious fitting to experimental data and fail to transfer across diverse chemistries, while highly accurate quantum-chemical calculations are far too computationally expensive for the large molecular systems and long timescales required to model polymer behaviors.

The article demonstrates that machine learning force fields (computational models trained on quantum-mechanical data to predict atomic forces rapidly) can accurately predict macroscopic polymer properties purely from first principles, without using experimental calibration data.

To accomplish this, the authors engineered Vivace, a scalable, local neural network architecture designed for large molecular dynamics simulations. They also developed PolyData, a training dataset containing hundreds of thousands of polymer structures labeled with quantum-chemical methods, and PolyArena, a benchmark comprising experimental bulk densities and glass transition temperatures across 130 diverse polymers. Vivace was trained on purely computational data and then evaluated by running molecular dynamics simulations to calculate real-world physical properties.

The analysis reveals several key findings. First, Vivace predicted polymer densities with a mean absolute error of 0.04 grams per cubic centimeter, outperforming established classical force fields (which exhibited errors between 0.07 and 0.10 grams per cubic centimeter) and matching or exceeding existing machine learning force fields. Second, the model generalized effectively to polymers unseen during training, yielding a density error of 0.06 grams per cubic centimeter. Third, Vivace successfully captured the thermal transition behavior of polymers, predicting glass transition temperatures across a test set with an average error of 43 Kelvin, outperforming baseline models. Finally, the authors found that incorporating periodic training data and maintaining an interaction cutoff radius of at least 6.5 ångströms were strictly necessary to prevent polymer chains from artificially repelling each other and collapsing the simulated bulk density.

These results establish that purely computational, quantum-derived machine learning models can bypass the extensive empirical parameterization historically required for polymer modeling. This capability enables rapid, reliable digital screening of novel polymer structures before initiating costly physical laboratory synthesis, reducing research and development risk across materials engineering programs.

Organizations developing next-generation materials should consider piloting machine learning force fields to accelerate screening pipelines for polymers. Further research and development should focus on extending this framework to predict mechanical properties such as elasticity, as well as modeling chemical reactivity for applications in polymer degradation, chemical recycling, and curing.

Readers should note specific limitations: machine learning force fields remain at least an order of magnitude slower than classical force fields, simulations currently do not explicitly model long-range electrostatic forces, and experimental reference measurements carry inherent variations based on processing history. Nevertheless, the evidence strongly supports that modern machine learning force fields provide high-fidelity macroscopic property predictions for non-reactive bulk polymers.

Microsoft neural networks SimPoly machine learning force fields polymer simulations

arXiv: 2510.13696 microsoft/simpoly

Paper: Machine Learning Force Fields, Oliver T. Unke et al. (2020). This comprehensive review lays the theoretical foundation, physical symmetry constraints, and training pipelines essential for constructing machine learning force fields from first-principles calculations.
Paper: Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics, Linfeng Zhang et al. (2017). It introduces deep potential molecular dynamics with quantum accuracy and linear scaling, establishing the core framework for simulating large-scale condensed matter systems with ab initio fidelity.
Paper: E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials, Simon Batzner et al. (2021). It details E(3)-equivariant graph neural networks for interatomic potentials, demonstrating how geometric equivariance dramatically enhances sample efficiency and accuracy in force-field modeling.
Paper: SchNet: A continuous-filter convolutional neural network for modeling quantum interactions, Kristof Schütt et al. (2017). It develops continuous-filter convolutional networks (SchNet) for energy conservation and analytic force prediction in molecular dynamics, a foundational milestone for neural force fields.
Paper: DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamics, Han Wang et al. (2017). It provides the software architecture and practical implementation methods for integrating deep learning potential energy models into standard molecular dynamics simulation engines.
Paper: Geometric and Physical Quantities improve E(3) Equivariant Message Passing, Johannes Brandstetter et al. (2022). It demonstrates how incorporating physical and geometric vector quantities into equivariant message passing improves predictive stability and accuracy for molecular physical interactions.
Paper: Neural Message Passing for Quantum Chemistry, Justin Gilmer et al. (2017). It establishes the foundational message passing neural network paradigm for quantum chemical property and energy prediction on molecular graphs.

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Cover for SimPoly: Simulation of Polymers with Machine Learning Force Fields Derived from First Principles

Abstract

Polymers are a versatile class of materials with widespread industrial applications. Advanced computational tools could revolutionize their design, but their complex, multi-scale nature poses significant modeling challenges. Conventional force fields often lack the accuracy and transferability required to capture the intricate interactions governing polymer behavior. Conversely, quantum-chemical methods are computationally prohibitive for the large systems and long timescales required to simulate relevant polymer phenomena. Here, we overcome these limitations with a machine learning force field (MLFF) approach. We demonstrate that macroscopic properties for a broad range of polymers can be predicted ab initio, without fitting to experimental data. Specifically, we develop a fast and scalable MLFF to accurately predict polymer densities, outperforming established classical force fields. Our MLFF also captures second-order phase transitions, enabling the prediction of glass transition temperatures. To accelerate progress in this domain, we introduce a benchmark of experimental bulk properties for 130 polymers and an accompanying quantum-chemical dataset. This work lays the foundation for a fully in silico design pipeline for next-generation polymeric materials.

Topics

neural networks

Keywords

SimPoly machine learning force fields polymer simulations machine learning interatomic potentials molecular dynamics simulation geometric molecular representation learning atomistic simulations amorphous polymers

1 Introduction
2 Results
2.1 Experimental Benchmarks and Computational Data for Polymer Simulations
2.2 Vivace – A Fast, Scalable, and Accurate MLFF
2.3 From Ab Initio Data to Bulk Properties with MLFFs
2.4 Observing Glass Transitions
3 Discussion
3.1 Modeling Inter-Molecular Interactions
3.2 Effect of the Cutoff Radius
4 Conclusions and Outlook
5 Methods
5.1 Details on Experimental Data
5.2 Classical Force Field Baselines
5.3 Machine-learning Force Field Baselines
5.4 Simulation Speed Measurements
5.5 MD Simulations for Density Calculations
5.6 Determination of the Glass Transition Temperature Through Simulation
5.7 Details on Computational Data
5.8 Quantum-Chemical Calculations
References
References
A Selected Polymers in PolyArena
B Model Details
B.1 Local environment construction
B.2 The initial embedding layer
B.2.1 Encoding and envelope function
B.2.2 Initialization of edge-related features
B.2.3 Initialization of node-related features
B.3 Mid-range invariant interaction
B.4 Short-range SE(3)-equivariant interaction
B.4.1 Tensor Product
B.5 Output layer
C Model Training
D Speed Measurements
E Additional Density and Glass Transition Figures
References
References

Knowls

Knowl 1 — Vivace Machine Learning Force Field Architecture
model/method
Vivace is a local $\mathrm{SE}(3)$ -equivariant graph neural network (GNN) interatomic potential designed for high-speed, large-scale molecular dynamics (MD) simulations of polymeric materials.

Vivace incorporates several key structural features:
1. Multi-Cutoff Interaction Hierarchy: Interatomic interactions are partitioned by distance into three regimes:
  - Short-Range Equivariant Regime ( $r_{ij} < r_c^S = 3.8\text{ \AA}$ ): Employs computationally intensive $\mathrm{SE}(3)$ -equivariant tensor products on spherical harmonics to capture direction-dependent covalent and strong short-range interactions.
  - Mid-Range Invariant Regime ( $r_c^S \le r_{ij} < r_c^M = 6.5\text{ \AA}$ ): Employs computationally lightweight invariant scalar operations and attention to capture weaker non-covalent (e.g., van der Waals) interactions.
  - Core Repulsion Modification ( $r_{ij} \le r_{\text{mod}} = 1.5\text{ \AA}$ ): Applies a polynomial repulsive energy modification to prevent atomic overlap without requiring an explicit Ziegler-Biersack-Littmark (ZBL) potential.
2. Strictly Local Receptive Field: Unlike conventional message-passing GNNs where information propagates over $K$ layers to yield an expanded receptive field of $K \cdot r_c$ , Vivace restricts all feature updates to the immediate 1-hop local neighborhood $\mathcal{N}_{r_c}(i) = \{j \mid r_{ij} \le r_c\}$ . The receptive field is strictly equal to the cutoff radius ( $6.5\text{ \AA}$ ), eliminating multi-GPU communication blocking across message-passing iterations and enabling efficient spatial domain decomposition.
3. Efficient Node-Level Tensor Products: Equivariant features are maintained and updated on nodes rather than on edges (as done in Allegro), reducing memory and computational scaling from $O(n_{\text{edge}})$ to $O(n_{\text{atom}})$ . Tensor products utilize channel-wise diagonal couplings scaling as $O(n_{\text{channel}} L_{\max})$ .
4. Linear-Complexity Three-Body Attention: Three-body interactions and triangular attention terms are evaluated via an inner product between updated equivariant node features and edge spherical harmonics, reducing the computational complexity from $O(M^2)$ to $O(M)$ for an atom with $M$ neighbors.
Knowl 2 — PolyArena Experimental Benchmark for Polymer Bulk Properties
experimental setup
PolyArena is a benchmark designed to evaluate machine learning force fields against experimental macroscopic bulk polymer measurements rather than computational reference energies and forces.

The benchmark consists of 130 chemically diverse polymers curated from the Bicerano handbook representing amorphous, atactic specimens synthesized and processed under standard bulk conditions:
- Elemental Scope: Contains main-group elements across the first three periods: H, C, N, O, F, Si, S, and Cl.
- Polymer Classes: Includes polyolefins, polyesters, polyethers, polyacrylates, polycarbonates, polyimides, polystyrenes, siloxanes, and perfluorinated polymers, bearing diverse functional groups (alkyl chains, nitriles, carboxylic acid derivatives, halogens).
- Molecular Weight: Repeating unit molecular weights span from $28\text{ g/mol}$ to $593\text{ g/mol}$ , ranging from simple polyolefins to complex AB-alternating copolymers.
- Volumetric Mass Densities: Standard-condition experimental densities span from $0.8\text{ g/cm}^3$ to $2.0\text{ g/cm}^3$ across all 130 polymers.
- Glass Transition Temperatures ( $T_g$ ): Experimentally measured $T_g$ values span from $152\text{ K}$ to $672\text{ K}$ , available for 87 of the polymers.
- Filtering Criteria: Filters out polymers with structural ambiguities, those failing Parameterized Consistent Force Field (PCFF) parameterization via Enhanced Monte Carlo, bromine-containing polymers incompatible with quantum basis sets, and computationally prohibitive structures (reducing the initial candidate set by approximately 10%).
Knowl 3 — PolyData Quantum-Chemical Dataset for Polymer MLFF Training
experimental setup
PolyData is a multi-modal quantum-chemical dataset specifically generated to train and fine-tune machine learning force fields for polymer systems. It consists of three complementary subsets:
1. PolyPack (~230,000 configurations):
  - Structure: Periodic unit cells containing two polymer chains of varying sizes (50, 100, and 250 atoms) initialized at densities spanning 0.1 to 1.4 times experimental density using Enhanced Monte Carlo (EMC). Structures undergo a 50 ps $NVT$ molecular dynamics trajectory at 1200 K, coordinate perturbation via Gaussian noise (standard deviation $\sigma = 0.08\text{ \AA}$ ), and 8 geometry optimization steps.
  - Quantum Method: CP2K/QUICKSTEP using mixed Gaussian and plane-wave basis sets (TZV2P Gaussian basis, 1200 Ry auxiliary plane-wave cutoff, norm-conserving GTH pseudopotentials), r2SCAN meta-GGA exchange-correlation functional, and Grimme D3 dispersion with Becke-Johnson damping (r2SCAN-D3(BJ)). Records energies, forces, and virial stress tensors. Probes dense intra-chain and short-range non-covalent interactions.
2. PolyDiss (~50,000 configurations):
  - Structure: Dissociation trajectories of isolated single polymer chains (3 repeating units) aligned along the $x$ -axis in an orthorhombic periodic box. The box is uniformly shrunk such that inter-image distances exceed $1.5\text{ \AA}$ , then expanded incrementally in steps of $0.1\text{ \AA}$ along $y$ , $z$ , $yz$ , and $xyz$ directions until inter-image separation reaches $7.0\text{ \AA}$ .
  - Quantum Method: CP2K r2SCAN-D3(BJ) single-point calculations saving energies, forces, and stress tensors. Probes inter-chain non-covalent binding and dissociation.
3. PolyCrop (~800,000 configurations):
  - Structure: Non-periodic spherical clusters extracted from packed periodic structures by cropping around a central atom. Only C–C single bonds crossing the sphere boundary are cleaved and capped with hydrogen atoms, preserving unsaturated bonds and local chemistry.
  - Quantum Method: ORCA calculations with range-separated hybrid DFT functional $\omega$ B97X-V and def2-TZVPP basis set (~400,000 clusters $\le 234$ atoms) and semi-empirical GFN2-xTB (~400,000 clusters $\le 1411$ atoms).
Knowl 4 — Vivace Equivariant Interaction and Three-Body Inner Product Mechanism
model/method
The short-range equivariant interaction in Vivace operates within a cutoff $r_c^S = 3.8\text{ \AA}$ . It updates equivariant node features $\tilde{\boldsymbol{\xi}}_i$ and invariant edge features $\mathbf{e}_{ij}$ using cross-attention and equivariant tensor products.
1. Equivariant Node Update: $\tilde{\boldsymbol{\xi}}_i^{(n+1)} = \tilde{\boldsymbol{\xi}}_i^{(n)} + \Delta \tilde{\boldsymbol{\xi}}_i$ $\Delta \tilde{\boldsymbol{\xi}}_i = \frac{1}{\text{Norm}_i} \tilde{\boldsymbol{\xi}}_i^{(n)} \otimes \sum_{j \in \mathcal{N}_S(i)} \mathbf{w}_{ij} \tilde{Y}_{ij}$ where $\tilde{Y}_{ij}$ are real spherical harmonics $Y_{lm}(\hat{\mathbf{r}}_{ij})$ computed from the normalized interatomic unit vector $\hat{\mathbf{r}}_{ij} = \mathbf{r}_{ij} / \|\mathbf{r}_{ij}\|$ , and $\text{Norm}_i$ is the channel-averaged $L_2$ norm of $\Delta \tilde{\boldsymbol{\xi}}_i$ over angular momenta $l$ and magnetic quantum numbers $m$ .
The edge weighting vector $\mathbf{w}_{ij}^{lp} \in \mathbb{R}^{n_{\text{channel}}}$ is obtained via invariant cross-attention: $\mathbf{w}_{ij}^{lp} = \text{Linear}_{lp}(\alpha_{ij} \mathbf{v}_{ij})$ $\alpha_{ij} = \exp\left( -\frac{\|\mathbf{q}_i - \mathbf{k}_{ij}\|_2^2}{2\sqrt{d}} \right) \eta_c\left( \frac{r_{ij}}{r_c^S} \right)$ where $\mathbf{q}_i = \text{Linear}(\mathbf{h}_i)$ , $\mathbf{k}_{ij} = \text{Linear}(\mathbf{e}_{ij})$ , $\mathbf{v}_{ij} = \text{Linear}(\mathbf{e}_{ij})$ , $d$ is the feature dimension, and $\eta_c$ is the smooth cutoff envelope.
1. Invariant Edge Update via Inner Product: $\Delta \mathbf{e}_{ij} = \text{Linear}\left( \Delta \tilde{\boldsymbol{\xi}}_i \cdot \tilde{Y}_{ij} \right) \odot \text{MLP}(\boldsymbol{\rho}_{ij})$ $\mathbf{e}_{ij}^{(n+1)} = \text{MLP}\left( \Delta \mathbf{e}_{ij} + \mathbf{e}_{ij}^{(n)} \right) + \mathbf{e}_{ij}^{(n)}$ where $\boldsymbol{\rho}_{ij} = \text{MLP}(\mathbf{b}_{ij}^S) \eta_c(r_{ij}/r_c^S)$ .
Expanding $\Delta \tilde{\boldsymbol{\xi}}_i$ yields the three-body equivalent formulation: $\Delta \mathbf{e}_{ij} \propto \sum_{k \in \mathcal{N}_S(i)} \left[ \left( \tilde{\boldsymbol{\xi}}_i \otimes \mathbf{w}_{ik} \tilde{Y}_{ik} \right) \cdot \tilde{Y}_{ij} \right] \odot \text{MLP}(\boldsymbol{\rho}_{ij}) = \sum_{k \in \mathcal{N}_S(i)} \tilde{\mathbf{A}}_{ik} \cdot \tilde{\mathbf{B}}_{ij}$ where $\tilde{\mathbf{A}}_{ik} = \tilde{\boldsymbol{\xi}}_i \otimes (\mathbf{w}_{ik}\tilde{Y}_{ik})$ and $\tilde{\mathbf{B}}_{ij} = \tilde{Y}_{ij} \odot \text{MLP}(\boldsymbol{\rho}_{ij})$ . This evaluation scales as $O(M)$ for an atom with $M$ neighbors rather than the standard $O(M^2)$ scaling of explicit three-body or triangular attention mechanisms.
1. Channel-Wise Factorized Tensor Product: For equivariant representations with angular momenta $l_1, l_2 \le L_{\max}$ , the tensor product weight tensor is factorized as $w_{n_1 n_2 n_3}^{l_3 \leftarrow l_1 l_2} = U_{n_1}^{l_1} V_{n_2}^{l_2} Q_{n_3}^{l_3}$ . Under channel-diagonal pairing ( $n_1 = n_2 = n_3 = n$ ), it simplifies to: $c_{n, l_3, m_3} = \sum_{l_3 \leftarrow l_1, l_2} \sum_{m_1, m_2} C_{(l_1 m_1),(l_2 m_2)}^{(l_3 m_3)} a_{n, l_1, m_1} b_{n, l_2, m_2}$ where $C_{(l_1 m_1),(l_2 m_2)}^{(l_3 m_3)}$ are Clebsch-Gordan coefficients, reducing weight scaling from $O(n_{\text{channel}}^3 L_{\max}^3)$ to $O(n_{\text{channel}} L_{\max})$ .
Knowl 5 — Vivace Energy, Forces, Stress, and Radial Envelopes
equation
The total potential energy $E$ predicted by Vivace is computed as the sum of atomic node energies, edge energies, short-range repulsive modifications, and atomic reference energies:

$E = \sum_i E_i^{\text{node}} + \sum_{i,j} \left( E_{ij}^{\text{edge}} + E_{ij}^{\text{mod}} \right) + \sum_i E_{Z_i}$

where:
- $E_i^{\text{node}} = \text{MLP}(\mathbf{h}_i^{\text{final}})$
- $E_{ij}^{\text{edge}} = \text{MLP}(\mathbf{e}_{ij}^{\text{final}})$
- $E_{Z_i}$ is an element-specific reference energy for atomic number $Z_i$
- $E_{ij}^{\text{mod}}$ is a short-range polynomial correction active for $r_{ij} \le r_{\text{mod}} = 1.5\text{ \AA}$ : $E_{ij}^{\text{mod}} = \sum_s \frac{1}{r_{ij}^s} \text{MLP}_s(\mathbf{e}_{ij}^{\text{final}}) \eta_c\left( \frac{r_{ij}}{r_{\text{mod}}} \right)$
Atomic forces $\mathbf{f}_i$ and the virial stress tensor $\boldsymbol{\sigma}$ are derived analytically: $\mathbf{f}_i = -\frac{\partial E}{\partial \mathbf{r}_i}, \quad \boldsymbol{\sigma} = \frac{1}{V} \frac{\partial E}{\partial \boldsymbol{\varepsilon}}$ where $\mathbf{r}_i \in \mathbb{R}^3$ is the nuclear position of atom $i$ , $\boldsymbol{\varepsilon} \in \mathbb{R}^{3 \times 3}$ is the lattice strain tensor, and $V \in \mathbb{R}$ is the unit cell volume.

Smooth Cutoff Envelope Function: For normalized distance $x = r_{ij} / r_c$ and smoothing width parameter $\delta = 2 / 6.5$ : $\eta_c(x) = \begin{cases} 1 & \text{if } 0 \le x \le 1 - \delta \\ 3(1-x)^2 - 2(1-x)^3 & \text{if } 1 - \delta < x < 1 \\ 0 & \text{if } x \ge 1 \end{cases}$

Exponential Radial Basis Functions: For basis index $k \in \{0, 1, \dots, n_{\text{basis}}-1\}$ : $B_k(r_{ij}) = \exp\left[ -\left( \frac{n_{\text{basis}}}{2(1 - e^{-r_c})} \right)^2 \left( e^{-r_{ij}} - \mu_k \right)^2 \right]$ $\mu_k = \frac{k}{n_{\text{basis}}} + \left( 1 - \frac{k}{n_{\text{basis}}} \right) e^{-r_c}$
Knowl 6 — 21-Step MD Relaxation Protocol for Polymer Packings
algorithm
The 21-step molecular dynamics relaxation protocol compresses and equilibrates amorphous polymer structures from an unphysical initial density (typically $0.5\text{ g/cm}^3$ , generated via Enhanced Monte Carlo with ~2000 atoms arranged as six equal chains) into realistic, fully relaxed bulk packings.

The protocol consists of 7 cycles, each composed of 3 successive simulation stages:
Input: Initial polymer configuration $X_0$ at density $\rho = 0.5\text{ g/cm}^3$ , target temperature $T_{\text{target}}$ , standard pressure $P_{\text{target}} = 1\text{ bar}$ , maximum temperature $T_{\max} = 700\text{ K}$ , maximum pressure $P_{\max} = 5 \times 10^4\text{ bar}$ , time step $\Delta t = 0.5\text{ fs}$
Output: Equilibrated polymer configuration $X_{\text{eq}}$ and relaxed unit cell volume $V_{\text{eq}}$
Initialize $X \leftarrow X_0$
for cycle $c = 1$ to 7 do
    Stage 1: Run $NVT$ MD at $T = T_{\max}$ with Nosé-Hoover thermostat damping $\tau = 50\text{ fs}$
    Stage 2: Run $NVT$ MD at $T = T_{\text{target}}$ with Nosé-Hoover thermostat damping $\tau = 50\text{ fs}$
    if $c \le 3$ then
         $P_c \leftarrow c \cdot (P_{\max} / 3)$
    else
         $P_c \leftarrow P_{\max} - (c - 3) \cdot ((P_{\max} - P_{\text{target}}) / 4)$
    end if
    Stage 3: Run $NPT$ MD at $T = T_{\text{target}}$ and $P = P_c$ with barostat damping $\tau = 500\text{ fs}$
end for
return Equilibrated state $(X, V)$
$T_{\max} = 700\text{ K}$ is set above the highest polymer glass transition temperature to ensure mobility and eliminate chain memory. $P_{\max} = 5 \times 10^4\text{ bar}$ forces the compaction of initially rigid polymer chains during the first 3 cycles, after which the pressure is released back to standard pressure ( $1\text{ bar}$ ) over the final 4 cycles.
Knowl 7 — Automated Hyperbolic Fitting and Bootstrapping Protocol for Tg Determination
algorithm
To identify polymer glass transition temperatures ( $T_g$ ) from MD trajectories without arbitrary manual thresholding, an automated ensemble simulation, inverse-variance aggregation, and bootstrapping protocol is applied.
Input: Polymer structure with $N = 10,000$ atoms, reference experimental $T_g^{\text{exp}}$ , number of independent replicas $R = 3$ , number of bootstrap samples $B = 1000$
Output: Estimated glass transition temperature $T_g^{\text{fit}}$ and uncertainty $\sigma_{T_g}$
Define temperature set $\mathcal{T} = \{T_g^{\text{exp}} - 160 + 20 \cdot k \mid k \in \{0, 1, \dots, 16\}\}$ , excluding $T < 100\text{ K}$
for each temperature $T \in \mathcal{T}$ do
    for each replica $r = 1$ to $R$ do
        Initialize box from independent packing $r$
        Run 21-step relaxation protocol at $T$
        Run $NPT$ production MD for $\ge 0.5\text{ ns}$ at $(T, P = 1\text{ bar})$
        Compute trajectory density mean $\rho_r(T)$ and variance $s_r^2(T)$
    end for
     $\bar{\rho}(T) \leftarrow \left( \sum_{r=1}^R \frac{\rho_r(T)}{s_r^2(T)} \right) / \left( \sum_{r=1}^R \frac{1}{s_r^2(T)} \right)$
     $\sigma^2(T) \leftarrow 1 / \left( \sum_{r=1}^R \frac{1}{s_r^2(T)} \right)$
end for
for bootstrap trial $b = 1$ to $B$ do
    For each $T \in \mathcal{T}$ , sample $\rho_b(T) \sim \mathcal{N}(\bar{\rho}(T), \sigma^2(T))$
    Fit hyperbolic transition function over $T \in \mathcal{T}$ :
         $\rho_b(T) = \rho_0 - a(T - T_g^{(b)}) - b_{\text{hyp}} \sqrt{(T - T_g^{(b)})^2 + c^2}$
    Extract fitted parameter $T_g^{(b)}$
end for
$T_g^{\text{fit}} \leftarrow \frac{1}{B} \sum_{b=1}^B T_g^{(b)}$
$\sigma_{T_g} \leftarrow \sqrt{\frac{1}{B-1} \sum_{b=1}^B \left( T_g^{(b)} - T_g^{\text{fit}} \right)^2}$
return $T_g^{\text{fit}}, \sigma_{T_g}$
Using large 10,000-atom systems is necessary to suppress finite-size density variance at high temperatures, reducing the fitting uncertainty of $T_g$ (e.g., from $\pm 50\text{ K}$ at 2,000 atoms to $\pm 22\text{ K}$ at 10,000 atoms for polystyrene).
Knowl 8 — Polymer Density Benchmark Results Across Force Fields
data/table
Mean absolute errors (MAEs) for standard-condition volumetric mass densities and MD simulation throughput were evaluated across classical force fields (PCFF, OPLS3e) and MLFFs (Vivace, UMA, MACE-OFF).

MD simulation speeds were measured on a single NVIDIA A100 GPU (80 GB) with a $0.5\text{ fs}$ time step using an ~8000-atom polystyrene system.
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- Density Accuracy: Vivace achieves the lowest overall density MAE ( $0.04\text{ g/cm}^3$ ) across the PolyArena benchmark, obtaining an MAE of $0.04\text{ g/cm}^3$ on training-seen polymers and $0.06\text{ g/cm}^3$ on 5 held-out unseen polymers (PVMS, PVC, PAN, PTPC, PODPM). UMA achieves an MAE of $0.06\text{ g/cm}^3$ across 10 selected polymers without having polymer data in its training set.
- Comparison to Classical FFs: Both Vivace ( $0.04\text{ g/cm}^3$ ) and UMA ( $0.06\text{ g/cm}^3$ ) outperform classical force fields PCFF ( $0.07\text{ g/cm}^3$ ) and OPLS3e ( $0.10\text{ g/cm}^3$ ), demonstrating that purely ab initio trained MLFFs predict bulk macroscopic polymer densities without empirical experimental parameterization.
- Simulation Speed: Vivace ( $0.52\text{ ns/d}$ ) and MACE-OFF ( $0.51\text{ ns/d}$ ) are roughly $17\times$ faster than UMA ( $0.03\text{ ns/d}$ ). Classical PCFF ( $17.68\text{ ns/d}$ ) is over an order of magnitude faster than all MLFFs.
- Failures: MACE-OFF failed on PCTFE simulations; PCFF failed due to numerical instabilities on PET and PODPM.
Knowl 9 — Glass Transition Temperature Prediction Performance Across Force Fields
empirical result
Glass transition temperatures ( $T_g$ ) were computed for 10 polymers (5 seen during Vivace training: PE, PIB, PET, PCTFE, PS; 5 unseen: PVMS, PVC, PAN, PTPC, PODPM) from density-temperature curves using 10,000-atom MD simulations.

Key results:
- Vivace: Achieves an overall $T_g$ $T_{g}$ MAE of $43\text{ K}$ $43 K$ across all 10 polymers, performing consistently on seen ( $43\text{ K}$ $43 K$ ) and unseen ( $44\text{ K}$ $44 K$ ) polymers. Specifically:
  - Polystyrene (PS): $T_g^{\text{fit}} = 399 \pm 22\text{ K}$ vs experimental $373\text{ K}$ (error $+26\text{ K}$ )
  - Polyethylene (PE): error $+36 \pm 26\text{ K}$
  - Polyisobutylene (PIB): error $+59 \pm 16\text{ K}$
  - Poly(ethylene terephthalate) (PET): error $+26 \pm 32\text{ K}$
  - Polychlorotrifluoroethylene (PCTFE): error $+68 \pm 21\text{ K}$
  - Poly[N,N'-(p,p'-oxydiphenylene)pyromellitimide] (PODPM): error $-44 \pm 38\text{ K}$
  - Poly(vinyl methyl sulfide) (PVMS): error $+1 \pm 106\text{ K}$
  - Poly(vinyl chloride) (PVC): error $+54 \pm 12\text{ K}$
  - Polyacrylonitrile (PAN): error $+90 \pm 121\text{ K}$
  - Poly[thio bis(4-phenyl)carbonate] (PTPC): error $+29 \pm 47\text{ K}$
- Baseline Comparisons:
  - PCFF achieves an MAE of $49\text{ K}$ , but failed completely on PET and PODPM due to simulation crashes.
  - MACE-OFF achieves an MAE of $62\text{ K}$ , failing completely on PCTFE.
  - OPLS3e achieves an MAE of $64\text{ K}$ .
- Systematic Bias: MLFFs (Vivace, MACE-OFF) systematically overestimate $T_g$ values, consistent with trends observed in classical force fields.
- Decoupling of Properties: An accurate room-temperature density prediction does not necessarily correlate with an accurate $T_g$ estimate, and vice versa.
Knowl 10 — Cutoff Radius Threshold and Inter-Chain Dissociation for Polymer Simulation Stability
empirical result
Analysis of force field cutoffs and training dataset composition reveals critical requirements for physical stability in polymer molecular dynamics:
1. Cutoff Radius Threshold for Density Stability: For polymers governed by non-covalent interactions (e.g., polychlorotrifluoroethylene, PCTFE), models with cutoff radii below $5.5\text{ \AA}$ fail to capture the inter-chain attractive well, predicting prematurely vanishing binding energies. In MD simulations, this causes polymer chains to unfold and repel, leading to catastrophic density collapse (simulated density approaching near-zero). A cutoff of $r_c \approx 6.5\text{ \AA}$ is required to capture the attractive minimum and maintain stable bulk densities. This explains why MACE-OFF ( $r_c = 4.5\text{ \AA}$ ) suffers density collapse on PCTFE, while UMA ( $r_c = 6.0\text{ \AA}$ ) and Vivace ( $r_c^M = 6.5\text{ \AA}$ ) remain stable.
2. Failure of Standard Static Computational Metrics to Predict MD Performance:
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Pre-trained Vivace (trained only on non-periodic OMol25 and PolyCrop clusters) achieved slightly lower force ( $179\text{ meV/\AA}$ ) and stress ( $1.7\text{ meV/\AA}^3$ ) MAEs on the PolyPack test set than fine-tuned Vivace ( $183\text{ meV/\AA}$ , $1.9\text{ meV/\AA}^3$ ), yet yielded a massive density MAE in MD simulations of $0.60\text{ g/cm}^3$ due to underestimating inter-chain dissociation energies. Fine-tuning on periodic inter-chain dissociation curves (PolyDiss) corrected the long-range potential well and reduced the MD density MAE to $0.04\text{ g/cm}^3$ .
Knowl 11 — Multi-GPU Scaling and Computational Throughput of Vivace
data/table
MD simulation throughput and parallel scaling across NVIDIA A100-SXM4 (80 GB) GPUs connected via NVLink were benchmarked using a $0.5\text{ fs}$ time step in LAMMPS via the ML-IAP KOKKOS interface.

Multi-GPU Scaling on a 15,552-atom Polystyrene System:
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Single-GPU Throughput Across Polystyrene System Sizes at Density $1.0\text{ g/cm}^3$ :
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- Scaling Mechanics: Vivace and MACE-OFF exhibit strong scaling up to 8 GPUs because their short interaction cutoffs ( $6.5\text{ \AA}$ and $4.5\text{ \AA}$ ) and high compute intensity result in computation dominating over communication. Conversely, classical PCFF throughput degrades beyond 2 GPUs ( $11.08\text{ ns/d} \to 8.22\text{ ns/d}$ at 8 GPUs) due to its larger cutoff ( $9\text{--}10\text{ \AA}$ ) and low computational overhead making inter-GPU communication the dominant bottleneck.

Coverage note — All primary contributions of the paper—including the Vivace architecture and multi-cutoff scheme, PolyArena benchmark, PolyData dataset, density prediction benchmarks, glass transition temperature estimation methodology and results, cutoff radius and inter-chain interaction stability analyses, and computational scaling measurements—are fully represented. Minor hyperparameter optimization ablation trials across unselected prototype configurations in Table S4 were omitted.

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Citation

MLA

Simm, G. N. C., et al. “SimPoly: Simulation of Polymers with Machine Learning Force Fields Derived from First Principles”. arXiv, 2025, http://arxiv.org/abs/2510.13696v1.

APA

Simm, G. N. C., Hélie, J., Schulz, H., Chen, Y., Simeon, G., Kuzina, A., Martinez-Baez, E., Gasparotto, P., Tocci, G., Chen, C., Li, Y., Cheng, L., Wang, Z., Nguyen, B. H., Smith, J. A., & Sun, L. (2025). SimPoly: Simulation of Polymers with Machine Learning Force Fields Derived from First Principles. arXiv. http://arxiv.org/abs/2510.13696v1

Chicago

Simm, G. N. C., J. Hélie, H. Schulz, et al. 2025. “SimPoly: Simulation of Polymers with Machine Learning Force Fields Derived from First Principles”. arXiv. http://arxiv.org/abs/2510.13696v1.

Harvard

Simm, G.N.C. et al. (2025) “SimPoly: Simulation of Polymers with Machine Learning Force Fields Derived from First Principles”, arXiv [Preprint]. Available at: http://arxiv.org/abs/2510.13696v1.

Vancouver

1. Simm GNC, Hélie J, Schulz H, et al (2025) SimPoly: Simulation of Polymers with Machine Learning Force Fields Derived from First Principles. arXiv

BibTeX

@article{simm2025simpoly,
  title = {SimPoly: Simulation of Polymers with Machine Learning Force Fields Derived from First Principles},
  author = {Simm, Gregor N. C. and Hélie, Jean and Schulz, Hannes and Chen, Yicheng and Simeon, Guillem and Kuzina, Anna and Martinez-Baez, Ernesto and Gasparotto, Piero and Tocci, Gabriele and Chen, Chi and Li, Yatao and Cheng, Lixue and Wang, Zun and Nguyen, Bichlien H. and Smith, Jake A. and Sun, Lixin},
  year = {2025},
  journal = {arXiv},
  url = {http://arxiv.org/abs/2510.13696v1},
  eprint = {2510.13696}
}

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SimPoly: Simulation of Polymers with Machine Learning Force Fields Derived from First Principles

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Knowls

Knowl 1 — Vivace Machine Learning Force Field Architecture

Knowl 2 — PolyArena Experimental Benchmark for Polymer Bulk Properties

Knowl 3 — PolyData Quantum-Chemical Dataset for Polymer MLFF Training

Knowl 4 — Vivace Equivariant Interaction and Three-Body Inner Product Mechanism

Knowl 5 — Vivace Energy, Forces, Stress, and Radial Envelopes

Knowl 6 — 21-Step MD Relaxation Protocol for Polymer Packings

Knowl 7 — Automated Hyperbolic Fitting and Bootstrapping Protocol for Tg Determination

Knowl 8 — Polymer Density Benchmark Results Across Force Fields

Knowl 9 — Glass Transition Temperature Prediction Performance Across Force Fields

Knowl 10 — Cutoff Radius Threshold and Inter-Chain Dissociation for Polymer Simulation Stability

Knowl 11 — Multi-GPU Scaling and Computational Throughput of Vivace

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