Molecular Dynamics (MD) simulations are crucial in materials science, but the accuracy versus efficiency trade-off of traditional methods limits their capabilities. *Ab initio* MD offers high accuracy but is computationally expensive, while classical force fields are efficient but less accurate. ML potentials, with their ability to represent multi-body interactions with unspecified functional forms, offer a potential solution to this dilemma. However, the accuracy of ML potentials is highly dependent on the quality and quantity of training data. The common approach is 'bottom-up' learning, using *ab initio* calculations (e.g., DFT) to generate training data. While more affordable than methods like CCSD(T), DFT calculations are still computationally expensive and may not accurately reflect experimental observations. This can lead to ML potentials that deviate from experimental measurements of key properties like lattice parameters and elastic constants. Addressing this issue requires strategies that either utilize large, diverse datasets from DFT calculations (potentially through active learning techniques) or incorporate experimental data directly. 'Top-down' learning, which utilizes experimental data, offers a way to circumvent the computational cost of generating *ab initio* data. However, experimental data is also scarce and often contains measurement errors. Furthermore, experimentally observable properties are ensemble averages, requiring simulations to compute these properties, which introduces complexities and computational challenges for backpropagation through the simulations. Methods like Differentiable Trajectory Reweighting (DiffTRe) offer a solution for time-independent properties, but models trained solely on limited experimental data can be under-constrained, leading to inconsistent results for off-target properties. This paper proposes a strategy that leverages both DFT calculations and experimental data to create more accurate ML potentials, building upon the existing concept of combining simulation and experimental data for classical force fields. It seeks to improve upon previous approaches by training a single deep ML potential instead of adding corrections to a DFT-trained potential.

Existing research on ML-based force fields highlights the challenges of data scarcity and the discrepancies between DFT simulations and experimental results. Several studies have explored bottom-up approaches using DFT calculations, but limitations in dataset size and the accuracy of DFT functionals have been consistently reported. Active learning strategies have been suggested to improve data efficiency, but robust uncertainty quantification remains a challenge. Top-down approaches, using experimental data, have also been investigated, but their limited data availability and the difficulty of performing backpropagation through long simulations restrict their applicability. Hybrid approaches combining DFT and experimental data have been suggested, often using a two-body correction applied to a DFT-trained ML potential. This approach, while improving certain properties, is limited in its ability to simultaneously reproduce multiple experimental observables and increases computational cost.

This study employs a Graph Neural Network (GNN) potential, specifically DimeNet++, implemented in JaxMD, to model titanium. The methodology involves a fused data training approach, combining DFT calculations and experimental data. The DFT dataset consists of 5704 samples, including various titanium structures (hcp, bcc, fcc) and configurations obtained through high-temperature MD simulations and active learning. The experimental dataset includes temperature-dependent (4-973 K) elastic constants and lattice parameters of hcp titanium, focusing on four key temperatures (23, 323, 623, and 923 K) to reduce computational burden. The training process utilizes two trainers: a DFT trainer and an EXP trainer. The DFT trainer performs standard regression on DFT-calculated energies, forces, and virial stress, using a weighted mean squared error loss function (Equation 1). The EXP trainer uses the DiffTRe method to optimize the parameters such that the properties computed from ML-driven MD simulations match experimental values (Equation 2). The gradients are calculated using the reweighting ansatz for the canonical ensemble (Equation 3). Three different training approaches are compared: (i) DFT pre-trained (only DFT trainer), (ii) DFT, EXP sequential (DFT trainer followed by EXP trainer), and (iii) DFT & EXP fused (alternating between DFT and EXP trainers). The latter two approaches are initialized with the parameters from the DFT pre-trained model, avoiding the need for prior potentials often used in top-down learning. All models are trained for a fixed number of epochs with early stopping, using appropriate numerical optimization hyperparameters. All MD simulations are performed in JaxMD, employing a velocity Verlet integrator, with appropriate thermostats and barostats used for different simulations (NVT and NPT) and properties.

The results demonstrate the superiority of the fused data training approach. The DFT pre-trained model achieves chemical accuracy in energy predictions (below 43 meV) but fails to reproduce experimental mechanical properties. The DFT, EXP sequential model shows a significant increase in energy error but accurately predicts mechanical properties, highlighting the importance of including DFT data. The DFT & EXP fused model provides a good compromise. It achieves a similar level of accuracy as the DFT pre-trained model in terms of energy, force, and virial predictions while accurately reproducing the experimental mechanical properties (bulk modulus, shear modulus, Poisson's ratio) within a few GPa. The errors for bulk modulus, shear modulus, and Poisson's ratio are below 3%, representing a significant improvement over the DFT-only model. The models also show good generalization to off-target properties. Phonon spectra are accurately predicted by all three models, with the DFT pre-trained model performing slightly better. Liquid-state properties (RDF, ADF, self-diffusion coefficients) are also well-reproduced, with models trained using experimental data exhibiting better agreement with experimental measurements. The generalization to different pressures also shows that the models accurately predict the lattice constants of hcp titanium under pressure, with DFT, EXP sequential model giving the best result. Evaluation of bcc titanium elastic constants reveals that the DFT & EXP fused model is the best-performing among the three trained models, based on comparison with the latest experimental data. The study highlights that while DFT data is essential for providing a reasonable energy landscape, the fusion of experimental data significantly improves the model's capacity to reproduce target experimental properties while not substantially harming the reproduction of off-target properties. The authors also showed that pre-training on DFT data can constrain the solution space and improve the reliability of the subsequent training on experimental data, a key finding that would need further investigation.

The findings address the research question of improving the accuracy of ML potentials by combining DFT and experimental data. The fused data training approach successfully corrects for DFT inaccuracies without significantly compromising the model's performance on off-target properties. This demonstrates that a single ML potential can be trained effectively using both simulation and experimental data. The superior performance of the combined approach highlights the importance of incorporating experimental constraints to improve the accuracy and reliability of ML potentials. The findings have broader implications for the development of accurate ML potentials for various materials and systems, particularly in cases where DFT calculations do not fully capture the experimental observations. This approach can facilitate more reliable MD simulations for predicting materials properties and accelerating materials discovery.

This study presents a successful strategy for generating highly accurate ML potentials by fusing DFT simulation and experimental data. The DFT & EXP fused model demonstrates significantly improved accuracy in predicting experimental mechanical properties compared to DFT-only models while maintaining reasonable performance on other properties. This approach is widely applicable and provides a valuable tool for materials modeling, promising improved accuracy and reliability in MD simulations for various materials and systems. Future work could investigate the effectiveness of different reweighting techniques and explore the optimal balance between DFT and experimental data in the training process.

The study primarily focuses on titanium, and the generalizability of the approach to other materials needs further investigation. Although the choice of four representative temperatures for experimental training yielded good temperature transferability, different temperature selections might show different results. The computational cost associated with the combined training approach remains relatively high and would become even higher for larger systems. The relative contribution of DFT and experimental data in the loss function can be tuned for different application requirements. Finally, the choice of the DiffTRe method is another limitation, as multi-state reweighting techniques could offer further improvements to the accuracy of ensemble averages.