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From Energy-Force Weighting to Primal-Dual Optimization of Machine-Learned Interatomic Potentials

This paper proposes a primal-dual optimization framework that replaces traditional energy-force weight selection with a force-constrained fitting procedure, demonstrating that this approach not only achieves comparable low-error performance with significant computational speedups across various materials but also provides a more robust and explicit rule for selecting machine-learned interatomic potentials.

Original authors: Chenyu Wang, Yangshuai Wang, Lei Zhang

Published 2026-10-02

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CC0 1.0

Original authors: Chenyu Wang, Yangshuai Wang, Lei Zhang

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

To understand how atoms stick together to form the world around us, scientists rely on a map called a potential-energy surface. This map describes the energy of a collection of atoms at every possible arrangement, acting as the rulebook for how they move, vibrate, and react. While the laws of quantum mechanics can calculate this map with perfect precision, doing so for every step of a simulation is far too slow for studying real materials. Instead, researchers use machine learning to build a shortcut: a model trained on a smaller set of perfect calculations that can predict the energy and forces for new arrangements almost instantly. These models are the engines behind modern simulations of everything from liquid water to molten salts.

However, teaching these models is a delicate balancing act. The training data provides two types of clues: the total energy of a system and the forces pushing or pulling on each atom. Energy tells the model which arrangements are stable, while forces dictate how the atoms will move. A model trained to be perfect at predicting energy might fail to capture the correct motion, and one focused too heavily on forces might lose track of stability. For years, scientists have tried to find the right mix by simply assigning a weight to each type of error, hoping that a specific number would produce the best model. But this approach treats the weight as a fixed setting, ignoring the fact that the meaning of that number changes depending on the entire training process used to create the model.

A new study by Chenyu Wang, Yangshuai Wang, and Lei Zhang challenges this traditional method. The researchers argue that the balance between energy and force is not a simple dial to be turned, but a complex choice that depends entirely on the specific protocol used to train the model. To test this, they examined three very different materials: molten lithium chloride, liquid water, and solid silicon. They first mapped out what happens when they train models using the old method of fixed weights. They found that the path of results was often messy and unpredictable. In some cases, increasing the weight for one type of error actually made the model worse at predicting both energy and forces, a phenomenon that defies simple intuition. They also discovered that the "best" model found by this method depended heavily on the specific computer algorithm used to solve the math, meaning there was no single, universal answer to be found by scanning through weights.

To solve this, the team replaced the search for the perfect weight with a different strategy: setting a strict limit on how much error is allowed in the force predictions. Instead of guessing a number, they told the computer, "Minimize the energy error, but do not let the force error exceed this specific threshold." The computer then adjusted the internal balance automatically to meet this rule. This approach, which uses a mathematical feedback loop to tune the model, proved to be far more efficient. In their tests, this new method reached the same high-quality results as the old method but did so six to nine times faster. It also proved more robust, finding the best models regardless of how the training was started.

The researchers then checked whether these different models actually produced different physical realities. They found that some properties were surprisingly stable; the arrangement of atoms in the immediate neighborhood of a central atom changed very little, regardless of how the model was balanced. However, other properties were much more sensitive. The way liquids flow and how solid materials respond to stress varied significantly depending on the specific balance chosen. This suggests that while the local structure of a material is forgiving, its behavior under motion or pressure is highly dependent on the precise details of the model.

The study concludes that there is no single "correct" machine-learned potential that works for every purpose. Instead, the model and the training protocol must be viewed as a single package. The choice of how to balance energy and force is not just a technical detail but a fundamental part of selecting the physical model itself. By shifting from a vague search for weights to a clear rule based on force limits, scientists can now select models more quickly and with greater certainty, ensuring that the simulations used to design new materials are built on a solid, explicit foundation.

Technical Summary: From Energy–Force Weighting to Primal–Dual Optimization of Machine-Learned Interatomic Potentials

Problem Statement
Machine-learned interatomic potentials (MLIPs) are typically trained by minimizing a weighted sum of energy and force errors, a process known as scalarization. This approach reformulates a multiobjective optimization problem into a single-objective one. However, the paper argues that the "nominal weight" used to balance these objectives is not an intrinsic physical parameter but is entirely dependent on the complete training protocol (including loss normalization, regularization, data partitioning, and optimization stopping rules). Consequently, a specific weight value identifies a potential only relative to that specific protocol.

Existing strategies either prescribe fixed coefficients or adapt them during optimization, but they do not resolve the fundamental issue of selecting a fitted potential under a specified protocol and a stated force-loss tolerance. Furthermore, empirical analysis reveals that the resulting energy–force trade-off paths (Pareto fronts) often contain dominated states at extreme weights and nonmonotone segments, making the selection of an optimal compromise difficult and solver-dependent.

Methodology
The authors propose replacing direct weight selection with a force-constrained model selection framework. Instead of scanning a grid of weights, the method minimizes the regularized energy objective subject to an upper bound on the normalized force loss.

Primal–Dual Formulation: The problem is cast as a constrained optimization where the Lagrange multiplier ( $\lambda$ ) acts as the force coefficient. The objective is to minimize $L_E(\theta) + R(\theta)$ subject to $L_F(\theta) \leq \varepsilon_F$ .
Adaptive Optimization: The authors implement a two-stage algorithm:
- Adaptive Stage: An iterative process alternating between primal descent (updating model parameters $\theta$ ) and projected dual ascent (updating the multiplier $\lambda$ based on the force-constraint residual). The multiplier is projected onto a bounded interval $[0, \Lambda_{max}]$ .
- Refinement Stage: Once the adaptive stage converges, the multiplier is frozen. A deterministic limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) algorithm is then used to refine the model parameters for the fixed $\lambda$ .
Fixed-Basis Context: The numerical experiments utilize fixed-basis Atomic Cluster Expansion (ACE) models. In this setting, the energy and force predictions are linear in the fitted coefficients, allowing for analytic gradients and reducing the problem to a regularized weighted least-squares subproblem during the refinement stage.

Key Results
The method was evaluated on three distinct chemical systems: molten LiCl (ionic), liquid H $_2$ O (molecular), and Si (covalent/solid-state).

Protocol Dependence: The empirical scalarization paths generated by independent L-BFGS fits at various weights were found to be protocol-dependent. They exhibited dominated states at extreme weights and, in the case of Si (which involved a distribution shift from crystalline training to amorphous testing), nonmonotone segments. The nondominated subsets varied depending on the solver used.
Performance of Constrained Method: The constrained primal–dual procedure successfully reached the solver-matched low-error region defined by the empirical Pareto subset.
- Across all three systems, the constrained fits remained within 2.42% of their solver-matched reference Root Mean Square Errors (RMSEs).
- The method achieved measured speedups of 6.7–8.9 times compared to completing full scalarization scans (which require independent fits for every weight).
- For LiCl, distinct warm starts converged to the same low-error region, suggesting the multiplier feedback reduces sensitivity to initialization.
Physical Observables: The study analyzed how the selected energy–force balance affects downstream physical properties:
- First-shell geometry: Found to be comparatively insensitive to the selected balance.
- Transport and solid-state observables: Showed stronger variation between models, indicating that the specific balance selected significantly impacts dynamic and structural response properties.

Significance and Claims
The paper claims that the fitted model–protocol pair should be regarded as the relevant object of energy–force model selection, rather than the weight value itself. By treating the energy–force coefficient as a dual variable associated with a stated force-loss tolerance, the proposed method provides an explicit selection rule that avoids the inefficiency and ambiguity of grid-based weight scanning.

The results support force-constrained fitting as a robust alternative to scalarization, offering a computationally efficient path to low-error models that are consistent with a specific physical tolerance. The study isolates the balance selection problem from representation learning by using fixed-basis ACE models, demonstrating that the proposed optimization strategy is effective even when the model architecture is held constant.

Technical Summary: From Energy–Force Weighting to Primal–Dual Optimization of Machine-Learned Interatomic Potentials

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