Upscaling DFT-trained machine-learning interatomic potential toward Quantum Monte Carlo accuracy: Sulfur-vacancy migration in monolayer MoS$_2$ as a testbed

This paper presents a multi-fidelity machine learning approach that fine-tunes a DFT-trained interatomic potential using limited quantum Monte Carlo energies to achieve near-QMC accuracy for simulating sulfur vacancy migration in monolayer MoS$_2$, enabling large-scale, high-precision simulations that are computationally prohibitive with direct QMC methods.

The Gist

Imagine you are trying to build a perfect map of a mountainous terrain to help hikers (atoms) navigate safely.

The Problem: The Map is Too Expensive or Too Rough
Scientists have two main ways to draw this map:

1. The "Good Enough" Map (DFT): This is like a standard GPS. It's fast, cheap to generate, and gives you a decent idea of where the hills and valleys are. However, it sometimes gets the height of the peaks wrong. If you are trying to cross a specific mountain pass (a chemical reaction), this map might tell you the pass is easy to climb when it's actually a steep cliff.
2. The "Perfect" Map (QMC): This is a satellite survey that measures every single rock and pebble with incredible precision. It gives the true height of the mountains. But, it is so expensive and slow to make that you can only afford to survey a tiny patch of land. You can't use it to map a whole continent or simulate a long hike because the computer would take centuries to finish.

The Solution: A Smart Hybrid Approach
The authors of this paper came up with a clever trick to get the best of both worlds. They wanted to upgrade their "Good Enough" map to be as accurate as the "Perfect" map, but without the impossible cost.

Here is how they did it, using a car tuning analogy:

• The Engine (The AI Model): They started with a car (an AI model called MACE) that was already built using the "Good Enough" map. This car drives well and knows how to handle turns (atomic forces) because it was trained on the fast, standard data.
• The Fuel Injection (The Energy Correction): They realized the car's speedometer (energy levels) was slightly off compared to the "Perfect" map. So, they took a few very expensive, high-precision fuel samples (QMC energies) from specific spots on the mountain.
• The Tuning (Fine-Tuning): Instead of rebuilding the whole car from scratch (which would be too hard), they only adjusted the dashboard and the speedometer (the "readout layers" of the AI). They used the expensive fuel samples to recalibrate the speedometer so it reads the true height of the mountains.
• The Safety Brake (Force Constraint): Here is the tricky part. If you just tweak the speedometer, the car might start driving wildly because the engine doesn't know how to handle the new speed. To prevent this, they added a "safety brake." They told the AI: "You can change the speed to match the perfect map, BUT you cannot change how the car steers (the forces) by more than a tiny, safe amount." This keeps the car stable and prevents it from crashing into imaginary cliffs.

The Test: Sulfur Vacancies in MoS2
To test this new method, they used a specific material: a thin sheet of Molybdenum Disulfide (MoS2). They looked at what happens when a single sulfur atom is missing (a "vacancy") and tries to move to a new spot. This movement is like a hiker trying to cross a ridge.

• The Old Way: The standard map said the hiker needed to climb a 2.30 eV hill.
• The Perfect Way: The expensive, high-precision survey said the hill was actually 2.85 eV. That's a huge difference!
• The New Hybrid Way: Their tuned model predicted 2.75 eV. It was almost as accurate as the expensive survey but calculated instantly.

The Results

• Accuracy: The new model got the energy barriers (the height of the hills) almost exactly right, matching the expensive "gold standard" results within a tiny margin of error.
• Forces: Even though they didn't use the expensive data to teach the model how to steer (forces), the "safety brake" kept the steering accurate. The model's steering became much better than the original, matching the high-precision survey almost as well as the original map did.
• Scale: Because the model is fast, they could simulate huge scenarios—like a whole line of missing atoms moving at once—that would have been impossible to calculate with the expensive method.

In Summary
The authors created a "smart upgrade" for computer simulations. They took a fast, slightly inaccurate model and gave it a tiny dose of expensive, high-precision data to fix its energy readings, while using a safety rule to keep its movement predictions stable. This allows scientists to run massive, high-accuracy simulations of materials that were previously too difficult or expensive to study.

Technical Summary

Technical Summary: Upscaling DFT-trained MLIPs toward QMC Accuracy

Problem Statement
Accurate modeling of potential-energy surfaces (PES) is critical for simulating activated processes like vacancy diffusion and phase transformations. While machine-learning interatomic potentials (MLIPs) enable large-scale sampling and free-energy calculations that are computationally prohibitive for first-principles methods, their accuracy is inherently limited by the reference data used for training. Standard Density Functional Theory (DFT)-trained MLIPs reproduce DFT results, which often contain systematic biases in barrier heights and defect energetics. Conversely, Quantum Monte Carlo (QMC) methods offer benchmark-quality energies approaching chemical accuracy but are currently too expensive for extensive sampling. Furthermore, obtaining converged atomic forces from stochastic QMC methods (specifically fixed-node diffusion Monte Carlo, FN-DMC) is significantly more difficult and less routine than calculating energies, creating a bottleneck for training high-fidelity MLIPs that rely on force data.

Methodology
The authors propose a multi-fidelity learning (MFL) strategy to "upscale" a DFT-trained MLIP to near-QMC accuracy without requiring direct QMC force calculations. The approach utilizes a partially frozen fine-tuning (FT) scheme on an equivariant message-passing neural network (MACE). The methodology consists of three core components:

1. Data Generation: A dataset of approximately $10^3$ configurations was generated using constrained molecular dynamics (MD) and nudged elastic band (NEB) paths based on a pre-existing DFT-trained MACE potential. Single-point FN-DMC energies were calculated for these configurations, while atomic forces were retained from the underlying DFT calculations.
2. Fine-Tuning Architecture: The authors fine-tuned only the "readout" layers of the MACE model, which map learned invariant features to per-atom energies. The equivariant message-passing layers, which encode the geometric representation of local environments and the DFT-learned force field, were frozen. This preserves the qualitative structural physics learned from DFT while allowing the energy mapping to be recalibrated to QMC targets.
3. Loss Function and Constraints: The training objective minimizes a combined loss function containing a mean squared error (MSE) term for the FN-DMC energies and a thresholded penalty term for the deviation of predicted forces from the DFT baseline forces.
   • The force penalty is defined as $FEt(F_{pred}, F_{DFT}) = \Theta(\|\Delta F\|^2 - t^2)(\|\Delta F\|^2 - t^2)^2$, where $t$ is a threshold parameter (set to 16 eV/Å).
   • This constraint prevents the model from developing unphysical forces or large deviations from the stable DFT dynamics while still allowing the energy landscape to shift toward QMC accuracy.

Key Contributions

• Force-Constrained Upscaling: The paper demonstrates a practical protocol to correct DFT-trained MLIPs using high-level QMC energies and low-level DFT forces, circumventing the need for expensive and noisy QMC force calculations.
• Partial Freezing Strategy: By freezing the message-passing layers and updating only the readout, the authors maintain the stability of the DFT force field while achieving QMC-level energetics.
• Multi-Fidelity Validation: The study validates that a limited dataset of QMC energies (as few as 37 samples) is sufficient to significantly improve the model, with performance stabilizing around 500 samples.

Results
The method was tested on sulfur (S) vacancy migration in monolayer MoS2, a system involving mono-, bi-, and quad-vacancies.

• Energetics: The fine-tuned MLIP (FT-MLIP) achieved near-QMC accuracy for migration barriers. For a mono-vacancy, the FT-MLIP barrier (2.75 eV) differed by only ~0.1 eV from the explicit FN-DMC result (2.85 eV), whereas the baseline DFT-MLIP was 0.55 eV lower.
• Forces: Although not trained on QMC forces, the FT-MLIP showed improved force fidelity. The mean absolute error (MAE) of atomic forces relative to QMC derivatives decreased from 220 meV/Å (DFT-MLIP) to 160 meV/Å (FT-MLIP).
• Generalization (Out-of-Domain): The model successfully predicted migration barriers for bi-vacancies and quad-vacancies (transfer tests) with deviations of only 0.04–0.15 eV from explicit FN-DMC calculations, significantly outperforming the DFT baseline.
• Free Energy: The approach enabled large-scale thermodynamic integration simulations to calculate free energy barriers at 300, 600, and 900 K, revealing that vibrational entropy corrections are comparable in magnitude to the QMC energy corrections and can qualitatively alter transition state locations.

Significance
The authors claim that this approach opens the window to large-scale, near-QMC quality simulations for systems and configurations (such as large supercells with multiple defects) that are inaccessible to direct brute-force QMC methods. The method provides a controlled trade-off between learning high-level energy corrections and preserving the qualitative stability of the DFT force field. The authors assert that the technique is generalizable to other systems where electronic correlation affects energetics but does not qualitatively alter the PES, offering a computationally bearable path to benchmark-quality materials simulations.