Ultra-Fast Force Field (UF3)

• UF3 is a machine-learned interatomic potential framework that employs spline-based basis expansions and rigorous regularization for precise energy and force predictions.
• It integrates two- and three-body interaction models to accurately simulate structural properties and epitaxial growth dynamics across diverse material systems.
• UF3 further incorporates quantum corrections via the Wigner–Kirkwood expansion and adapts to applications from nuclear quantum effects to fast solar magnetic field fitting.

The Ultra-Fast Force Field (UF3) is a class of machine-learned interatomic potentials and effective force field transformations for atomistic simulation, characterized by its explicit construction using spline-based basis expansions, rigorous regularization, and computational efficiency rivaling empirical potentials. UF3 methods underpin both modern MLIP strategies in materials science and advanced quantum-corrected effective potentials for including nuclear quantum effects. The UF3 methodology has demonstrated high-fidelity prediction of energies, forces, and structural properties in a range of material systems—including Si–N and AlN—at computational cost accessible for large-scale molecular dynamics (MD) and epitaxial-growth simulations (Gibson et al., 11 Sep 2024, Taormina et al., 11 Nov 2025, Sundar et al., 2017). Multiple independent research strands have converged under the UF3 nomenclature, spanning machine-learned spline potentials, force-field functor constructions, and fast analytic field-fitting schemes in solar physics, all united by the principle of combining physically motivated, interpretable functional forms with optimized high-throughput training pipelines.

1. Mathematical Formulation and Theoretical Underpinnings

UF3 formulations in the domain of machine-learned interatomic potentials represent the total energy as a sum over two- and three-body interactions, each expanded in a compact, interpretable B-spline basis.

For a system with $N_s$ atoms: $$E_{\text{total}}(\{r_i\}) = \sum_{i=1}^{N_s} \sum_{j \in \textrm{nbr}(i)} V_2(r_{ij}) + \sum_{i=1}^{N_s} \sum_{j \in \textrm{nbr}(i)} \sum_{k > j,\,k\in \textrm{nbr}(i)} V_3(r_{ij}, r_{ik}, r_{jk})$$ where $V_2(r)$ and $V_3(r_{ij}, r_{ik}, r_{jk})$ are expanded as: $V_2(r) = \sum_{n=0}^{K} c_n B_n(r)$

$$V_3(r_{ij}, r_{ik}, r_{jk}) = \sum_{l=0}^{K_l} \sum_{m=0}^{K_m} \sum_{n=0}^{K_n} c_{lmn} B_l(r_{ij}) B_m(r_{ik}) B_n(r_{jk})$$

with $B_n(r)$ denoting cubic B-spline basis functions and $c_n$, $c_{lmn}$ the coefficients determined via linear regression. Only pair distances and triplets are used as descriptors, eschewing explicit higher-order angular or atom-centered invariants (Gibson et al., 11 Sep 2024, Taormina et al., 11 Nov 2025).

The force on atom $i$ is computed analytically from the derivative of the energy with respect to atomic coordinates, reflecting a tight coupling between potential smoothness and force fidelity.

In the quantum-corrected regime, UF3 (also referred to as the force-field functor FFF) constructs an effective potential via the Wigner–Kirkwood expansion: $$\widetilde{V}(\mathbf{r}) = V(\mathbf{r}) + \frac{\hbar^2}{24 m} \left[ \beta \nabla^2 V(\mathbf{r}) - \beta^2 |\nabla V(\mathbf{r})|^2 \right]$$ where $V(\mathbf{r})$ is the classical potential, $\beta = 1/(k_B T)$, and $m$ is the atomic mass. This approach allows for quantum equilibrium distributions at near-classical simulation cost (Sundar et al., 2017).

Regularization in UF3 employs both $L_2$ (ridge) and smoothness (curvature) penalties: $$L(c) = \sum_{s \in \text{train}} \left[ (E^{s}_{\text{pred}}(c) - E^{s}_{\text{ref}})^2 + w_f \| F^{s}_{\text{pred}}(c) - F^{s}_{\text{ref}} \|^2 \right] + \alpha_2 \| c^{(2)} \|^2 + \beta_2 \| D c^{(2)} \|^2 + \alpha_3 \| c^{(3)} \|^2 + \beta_3 \| D c^{(3)} \|^2$$ (Gibson et al., 11 Sep 2024, Taormina et al., 11 Nov 2025). This ensures stability and transfers directly to interpretable, physical spline shapes.

2. Training Data Generation, Selection, and Ablation

UF3's predictive accuracy and transferability are contingent upon the diversity and quality of the training dataset. Datasets typically combine:

• Expert-curated configurations: Ab initio MD trajectories, high-pressure/high-temperature relaxed structures, and defect-containing cells in representative crystal structures (e.g., $\alpha$, $\beta$, $\gamma$ Si$_3$N$_4$, wurtzite/rock-salt AlN) (Gibson et al., 11 Sep 2024, Taormina et al., 11 Nov 2025).
• Autonomously generated structures: Genetic algorithms (e.g., GASP) produce a broad coverage of non-equilibrium, compositional, and configurational diversity, encompassing metastable and high-energy motifs.

Filtering is applied to exclude structures with extreme energies (e.g., $>$3–4 eV/atom above the convex hull), excessive atomic forces, or unwanted chemical motifs (e.g., N-rich clusters or N$_2$ dimers), as these degrade both energy/force regression and dynamical properties if included indiscriminately (Gibson et al., 11 Sep 2024, Taormina et al., 11 Nov 2025).

A critical contribution is the identification of "diversity-induced underfitting": when the training set spans configurations beyond the model's representational capacity, the result is degraded performance—even relative to models trained on smaller, more targeted datasets. An ablation workflow tests variant models on incrementally filtered datasets, assessing target-property regression, structural predictions (e.g., RDF, density), and equilibrium/dynamical properties to identify optimal diversity thresholds (Gibson et al., 11 Sep 2024).

3. Model Fitting, Regularization, and Hyperparameter Optimization

UF3 models are fit via regularized linear regression, obviating the need for stochastic gradient descent or neural networks. Featurized structures (distances, triplets) are precomputed and stored, enabling rapid retraining for hyperparameter sweeps.

Typical hyperparameters include:

• Number and location of B-spline knots (controls basis complexity),
• Regularization strengths: $\alpha_{2,3}$ (ridge), $\beta_{2,3}$ (curvature),
• Radial cutoffs for 2- and 3-body interactions (e.g., $R_c\approx6$ Å for AlN, $R_2=6$ Å, $R_3=4$ Å for SiN),
• Force-to-energy loss weighting ($w_F$, $w_E$),
• Dataset subset weights for application tuning (Taormina et al., 11 Nov 2025, Gibson et al., 11 Sep 2024).

Hyperparameter selection typically employs automated search frameworks (Optuna TPE) with cross-validation against reference DFT benchmarks—lattice constants, elastic constants, defect formation energies, and simulated surface energies—prior to MD or application-specific testing (Taormina et al., 11 Nov 2025).

A distinctive feature is the use of analytic and interpretable sums over few parameters, which makes UF3 coefficients physically meaningful and ensures that overfitting tendencies can be detected and manually addressed via loss term weighing or spline-knot adjustment.

4. Performance Metrics and Validation

UF3 approaches, when properly trained and regularized, yield:

• Force and energy RMSE on held-out DFT data typically in the range 0.02–0.2 eV/atom and 0.007–0.2 eV/Å (system-specific, stronger for near-equilibrium structures) (Taormina et al., 11 Nov 2025).
• Elastic constants in SiN and AlN predicted within $<$10–20% of DFT for most components; certain challenging constants (e.g., $C_{13}, C_{33}$ in AlN) underpredicted by $\sim 20$–$25$\% (Taormina et al., 11 Nov 2025).
• Accurate surface energies: e.g., [0001] AlN surface energy within $-3.6\%$ of DFT (Taormina et al., 11 Nov 2025).
• Structure prediction: Amorphous and crystalline densities, RDFs, and ADFs match ab-initio benchmarks within their measurement uncertainty—demonstrated for a-Si$_3$N$_4$ and AlN epitaxial layers (Gibson et al., 11 Sep 2024, Taormina et al., 11 Nov 2025).
• Defect and dislocation core fidelity: Reproduction of known high-resolution atomic motifs, including failed predictions by non-UF3 models (e.g., correct 8-atom ring for edge dislocation core, not spurious 4-atom ring) (Taormina et al., 11 Nov 2025).
• Computational cost: UF3 achieves 9,000–10,000$\times$ speedup over DFT MD, frequently surpasses neural-network MLIPs by $\sim$8–10$\times$, and approaches the cost of traditional empirical force fields, lagging only by a factor of 2–10$\times$ depending on details of cutoffs and descriptor computation (Gibson et al., 11 Sep 2024, Taormina et al., 11 Nov 2025).

For quantum-corrected UF3/FFF, RDF peak shifts, densities, and coexistence curve properties reproduce RPMD and experiment within $1$–$2\%$, at $100\times$ reduced computational cost compared to path-integral methods (Sundar et al., 2017).

5. Application Domains: Epitaxial Growth, Quantum Effects, and Magnetic Field Fitting

Atomistic Epitaxy and Complex Material Modeling

The UF3 AlN potential accurately reproduces experimental and DFT phase stability, mechanical properties, and growth dynamics in large-scale epitaxial-growth simulations (up to 200,000 atoms, nanosecond time scales). It captures both the correct wurtzite overlayer crystallography and experimentally-observed layer-by-layer (Frank–van der Merwe) growth, outperforming traditional Stillinger–Weber and other MLIPs that fail to stabilize the correct polytype or yield mixed-phase growth with unphysical voids (Taormina et al., 11 Nov 2025).

Inclusion of Nuclear Quantum Effects

Through the FFF construction, UF3 modifies classical potentials via the Wigner–Kirkwood expansion to account for nuclear quantum fluctuations, yielding effective forces that allow reproduction of quantum equilibrium properties (such as quantum-delocalized RDFs and accurate coexistence properties in Ne) in classical MD codes with only minor additional computational overhead (Sundar et al., 2017). This approach is extensible to flexible biomolecular force fields and water models.

Fast Forward-Fitting of Magnetic Fields

In solar physics, UF3 refers to a rapid, second-order analytic forward-fitting code for nonlinear force-free magnetic field extrapolation in the solar corona. The method efficiently recovers the force-free $\alpha$ parameter and field-line topology by superposing second-order solutions for twisted flux tubes, fitted directly to stereo-triangulated coronal loop observations and vector magnetograms (Aschwanden et al., 2012). This enables recovery of non-potential field structure in seconds to minutes, far faster than optimization-based NLFFF codes.

6. Limitations, Pitfalls, and Future Directions

• Model Complexity vs. Data Diversity: Excessive data diversity beyond the model's basis capacity leads to degraded regression and dynamical properties (“diversity-induced underfitting”). An ablation-driven workflow and basis size scaling are essential for dataset selection (Gibson et al., 11 Sep 2024).
• Elastic Constant Limitations: Underprediction of certain properties (e.g., $C_{13}, C_{33}$ in AlN) reflects the intrinsic limits of the current basis or dataset. Increasing spline-knot resolution or adaptive knot placement may ameliorate these errors (Taormina et al., 11 Nov 2025).
• Linear Fitting Constraints: The analytic, linear nature cannot capture all many-body PES subtleties attainable by deep NNs or GNNs. Targeting application-specific accuracy sometimes requires trade-offs against less critical metrics.
• Quantum Corrections: Wigner–Kirkwood-based UF3 becomes less accurate when higher-order $\hbar^4$ terms matter (cryogenic $T$ or light atoms); inclusion of these terms or many-body Wigner corrections is an open area (Sundar et al., 2017).
• Magnetic Field Fitting: Only uniformly-twisted, closed flux tubes are represented; arbitrary current sheets or strongly kink-unstable regions are not captured in solar UF3 (Aschwanden et al., 2012).

Planned improvements include GPU-accelerated descriptor computation, multi-component and multi-scale extensions (e.g., for Ga–Al–N and heterostructures), hybrid atomistic–continuum integration, and ML surrogate modeling for the quantum correction term.

7. Summary Table: Key Features and Benchmarks

Aspect	UF3 MLIP (SiN, AlN)	FFF/Quantum UF3	Solar UF3
Energy Model	Spline-based 2+3-body sums	Classical + $\hbar^2$ correction	Twisted flux-tube analytic
Fitting	Linear regression, regularized	Analytic (Wigner–Kirkwood)	Analytic superposition fit
Training Data	DFT + MD + GA, filtered	Any (suitable $V$)	Magnetograms + loop constraints
Accuracy	$\sim$0.01–0.2 eV/atom MAE	Quant. obs. within 1–2%	$\alpha_{\text{fit}}$ to 70–100%
Speed	$>$9,000$\times$ DFT, $\sim$2–5$\times$ slower than finest empirical	1.1$\times$ classical MD	Seconds–minutes per run
Application Domain	Large-scale MD, epitaxy, defects	NQE in MD (Ne, water, bio)	Solar coronal fields

• "When More Data Hurts: Optimizing Data Coverage While Mitigating Diversity Induced Underfitting in an Ultra-Fast Machine-Learned Potential" (Gibson et al., 11 Sep 2024)
• "Machine-learning interatomic potential for AlN for epitaxial simulation" (Taormina et al., 11 Nov 2025)
• "Reproducing Quantum Probability Distributions at the Speed of Classical Dynamics: A New Approach for Developing Force-Field Functors" (Sundar et al., 2017)
• "A Nonlinear Force-Free Magnetic Field Approximation Suitable for Fast Forward-Fitting to Coronal Loops. II. Numeric Code and Tests" (Aschwanden et al., 2012)