All DFT calculations, including structural relaxations and single-point calculations, were carried out using VASP with the projector-augmented-wave (PAW) method^[17-20]. A plane-wave cutoff of 520 eV was used. Input files were generated using the PBE pseudopotentials through pymatgen^[21] MPRelaxSet (POT_GGA_PAW_PBE, corresponding to the 06-05-2010 PBE PAW dataset in VASP documentation). Specifically, the Ni_pv potential (PAW_PBE Ni_pv 06Sep2000) was used with 3p⁶3d⁸4s² electrons treated as valence, together with the standard O potential (PAW_PBE O 08Apr2002) with 2s²2p⁴ electrons treated as valence. For the interpolated unit-cell configurations, self-consistent (SCF) static calculations were first carried out on a Γ-centered 8 × 8 × 5 k-point grid to obtain the energies, magnetic moments, and charge densities. Subsequent non-self-consistent (NSCF) calculations reused the converged charge densities and employed a denser Γ-centered 16 × 16 × 10 k-point grid to obtain the projected density of states (PDOS) in Supplementary Figure 6. For all ab initio molecular dynamics (AIMD) calculations, Γ-centered k-point grids of 4 × 4 × 2 and 3 × 5 × 2 were used for the Oct and the Sqr phases, respectively. Gaussian smearing with ISMEAR = 0 and SIGMA = 0.05 eV was employed. Electronic convergence (EDIFF) was set to 1 × 10^-6 eV, and structural relaxations were converged to a force threshold of 0.02 eV/Å. For all DFT+U calculations, we consistently employed the rotationally invariant form introduced by Dudarev et al.^[22] with U_eff = U - J = 6.2 eV applied to the Ni 3d orbitals, following the convention implemented in pymatgen. For all spin-polarized calculations, including the interpolated static calculations and AIMD simulations, the initial magnetic moment of O was set to +0.01 μ_B, while the magnitude of the initial Ni magnetic moment was set to 3.0 μ_B. The signs of the Ni moments were assigned according to the FM or antiferromagnetic (AFM) initialization protocol. Throughout this work, we use “FM-initialized” and “AFM-initialized” to refer to the imposed electronic/magnetic initialization rather than to guarantee uniform convergence to ferromagnetic or antiferromagnetic states, as the SCF magnetic solutions can be more complex, particularly in the Sqr region.

In order to compare Oct–Sqr energetics under different correlation treatments, we employed PBE^[23,24], PBE+U, r2SCAN^[25], and HSE06^[26]. No additional Hubbard correction was applied to r2SCAN and HSE06. In these computations, we used triclinic cells containing two NiO formula units [Figure 1, lower panels]. The Oct and Sqr endpoint structures were first relaxed using non-spin-polarized PBE without U. Intermediate configurations were then generated by linearly interpolating both the atomic coordinates and lattice matrices. The resulting fixed interpolation pathway was evaluated using single-point calculations at each electronic-structure level.

To obtain PBE and FM-initialized PBE+U datasets for fine-tuning MLFFs, we performed AIMD in the isothermal–isobaric (NPT) ensemble at T = 300, 600, 900 K and P = 1 bar. Both the cell volume and the cell shape were allowed to vary. The AIMD trajectories were propagated using the Langevin thermostat and barostat, with lattice friction coefficients of 10 ps^-1 and a fictitious lattice mass of 1,000 atomic mass units. At each temperature, we performed two simulations, initialized from the Oct and the Sqr structures, respectively. Each simulation lasted 250 steps with a time step of 4.0 fs. From the resulting PBE trajectories of the Oct and the Sqr phases, we extracted 750 electronically converged frames for each phase. From the FM-initialized PBE+U trajectories, we extracted 750 and 250 electronically converged frames for the Oct and the Sqr phases, respectively. For Sqr-NiO, only 250 frames in total across the three PBE+U trajectories reached electronic self-consistency; the remaining unconverged frames were excluded, as detailed in Supplementary Section 4.4. These extracted frames subsequently formed the datasets for fine-tuning and validating MLFFs. For AIMD, the two-formula-unit cells were replicated 2 × 2 × 2, yielding supercells containing 16 NiO formula units to sample a broader range of local configurations.

Representative INCAR files used for the FM-initialized PBE+U fixed-path static calculations and 300 K AIMD simulations, along with detailed representative VASP settings, are provided in Supplementary Section 4.

For the datasets generated from AIMD calculations, we first held out 10% of all configurations at random for testing. Training configurations were then selected from the remaining configurations by maximizing the information entropy implemented in QUESTS^[27]. The information entropy of a set of n structures (H) is defined as^[28]

(1) $$ H(\mathrm{\mathbf{\{X\}} })=-\frac{1}{n}\sum_{i=1}^n\mathrm{log}\left [\frac{1}{n} \sum_{j=1}^n K_{\mathrm{h}}(\mathrm{\mathbf{X} }_i,\mathrm{\mathbf{X} }_j)\right ] , $$

where X_i denotes the vectorized local atomic features of structure i, K_h is a Gaussian kernel, and h is the kernel bandwidth, set to h = 0.015 in this work. The local atomic features were constructed using 32 channels and a radial cutoff of 5.0 Å. The differential entropy gain upon adding a candidate structure Y into the existing dataset {X} is given by

(2) $$ \delta H(\mathrm{\mathbf{Y} }|\mathrm{\mathbf{\{X\}} })=-\mathrm{log}\left [\sum_{i=1}^n K_{\mathrm{h}}(\mathrm{\mathbf{Y} },\mathrm{\mathbf{X} }_i)\right ] . $$

All DFT-labeled structures were ranked by δH, and the top N_train configurations were selected to maximize the information diversity of the training set for a given training-set size.

Using this approach, we selected N_train = 660 configurations from the PBE dataset to fit the pretrained MLFFs accurately to the non-spin-polarized PBE surface. For the PBE+U dataset, we varied N_train from 50 to 290 in increments of 40 to investigate how rapidly test errors decrease with increasing training-set size. Three independent train/validation–test splits were first generated for the PBE+U dataset. Within each split, the test set was held fixed across all training-set sizes, while progressively larger training subsets were selected from the corresponding train/validation pool by information-entropy maximization. The configurations not selected for training within the train/validation pool served as the validation set and were used only for on-the-fly monitoring of the validation loss during training with deepmd-kit. They did not contribute to gradient updates and were not used for final model evaluation, checkpoint selection, or hyperparameter optimization; all reported performance metrics were calculated exclusively on the held-out test set. The same matched data partitions were used for direct fine-tuning, sequential adaptation, and training-from-scratch comparisons.

Here, N_train denotes the number of configurations used for adaptation and therefore measures target-label efficiency within the fixed candidate pool generated in this study. Thus, N_train measures training-sample efficiency within the fixed 1,000-configuration PBE+U candidate dataset and should not be interpreted as either the total number of DFT+U calculations or the computational cost required to generate this candidate pool. In practical workflows, the candidate configurations need not be generated by full DFT-AIMD trajectories and may instead be obtained using lower-cost sampling such as pretrained MLFF MD followed by selective target-protocol labeling.

We compared four foundation MLFFs: the MPtraj branch and the OMat24 branch of DPA-3.1-3M^[15,29], denoted as DPA3-MP and DPA3-OMat, respectively, and two DPA4^[16] models, denoted as DPA4-Air-MP and DPA4-Neo-OMat. The neighbor cutoff radii were set to 6 Å for all these models. Models were fine-tuned and evaluated using deepmd-kit^[30]. For the training-from-scratch baseline, we used the DPA4-Neo architecture as a representative controlled baseline because it is the principal DPA-4 architecture examined in this work. The model was randomly initialized and trained using the same training-set sizes, matched data splits, learning-rate schedule, batch settings, loss definition, and other training settings used for DPA4-Neo fine-tuning. For direct fine-tuning, second-stage adaptation, and training from scratch, one training run with a fixed random seed was performed for each combination of data split and N_train; no additional repetitions with different optimization seeds were performed.

Models were trained for approximately 120 epochs, corresponding to ~120 × N_train/N_batch steps. Automatic batch sizing was used with a target of 128 atoms per batch. The gradient norm was clipped to a maximum of 1.0. For DPA-3, the learning rate decayed exponentially from 0.001 to 3 × 10^-5 over the full training schedule at an interval of N_train steps. The DPA-3 loss function combined energy, force, and virial mean squared error (MSE) terms. Their weights were varied from 0.2:100:0.02 at the beginning of training to 20:60:1 at the end. For DPA-4, the learning rate followed a warmup-stable-decay (WSD) schedule, starting at 1.4 × 10^-4, warming up to 7.0 × 10^-4 in N_train steps, remaining stable until 65% of the total training steps had been completed, after which it decayed cosinusoidally to 1 × 10^-6. The DPA-4 loss combined energy, force, and virial mean absolute error (MAE) terms with fixed weights of 1:1:0.25, respectively.

For the test-set energy-offset-aligned zero-shot RMSE, a uniform per-atom energy shift was applied to each pretrained model before calculating the RMSE. The shift was defined as the difference between the mean DFT energy per atom and the mean MLFF-predicted energy per atom over the corresponding test set, and the same constant was then added to all MLFF energy predictions in that test set. This removes arbitrary differences in the absolute energy references of pretrained MLFFs while preserving all relative energy differences among configurations. This aligned quantity is used only as a descriptive zero-shot baseline and is not used for model selection, hyperparameter optimization or quantitative claims regarding the recovery of relative energetics.

The MLFFs considered in this work are local, energy-conserving neural-network potentials that approximate the potential energy as a function of atomic species and geometry. For a configuration with atomic species Z_i and coordinates r_i, the total energy is expressed as a sum of atomwise contributions determined by the local environment N_i within a finite cutoff radius,

(3) $$ E(\{Z_i, \boldsymbol{r}_i\})=\sum_i \varepsilon _i(N_i). $$

Forces and virials are subsequently obtained from derivatives of the predicted total energy with respect to atomic coordinates and cell degrees of freedom. Consequently, these models represent a single-valued mapping from atomic geometry to energy and do not contain explicit electronic or magnetic state variables. This feature is particularly relevant to the present study, where different electronic-structure protocols may assign distinct energetic landscapes to closely related atomic configurations.

Here, we compare two generations of the Deep Potential architecture. DPA-3 is a multilayer graph neural network (GNN) constructed on a line graph series (LiGS), in which atoms, bonds, and angular relationships are represented through coupled graph levels and updated by message passing^[15]. The DPA-3.1-3M architecture used here employs an order-2 LiGS with 16 interaction layers and contains approximately 3.27 million trainable parameters. We evaluate two released DPA-3.1-3M checkpoints, denoted DPA3-MP and DPA3-OMat, corresponding to models pretrained on MP and OMat24 data, respectively.

DPA-4 is a later SE(3)-equivariant architecture based on edge-conditioned multi-focus attention (EMFA) SO(2)-equivariant convolution^[16]. It propagates scalar and higher-order angular features through equivariant message passing and uses the resulting scalar representation for atomwise energy prediction. In this work, we use two released DPA-4 variants: DPA4-Neo-OMat and DPA4-Air-MP. DPA4-Neo contains approximately 1.60 million trainable parameters, whereas the DPA4-Air architecture is larger and contains approximately 2.76 million parameters. The corresponding checkpoints were pretrained on data derived from OMat24 and MP, respectively. These models differ not only in pretraining history but also in architecture and model capacity, providing a useful set of pretrained priors for comparing downstream target-protocol adaptability under the same NiO fine-tuning task.

Readers are referred to the original model publications for further architectural details^[15,16].

To inspect how different electronic-structure treatments alter the NiO energy landscape and whether non-spin-polarized PBE and PBE+U provide a sufficiently challenging testbed for MLFF adaptability, we first relaxed the Oct and Sqr endpoint structures using non-spin-polarized PBE and generated intermediate configurations by linearly interpolating both the atomic coordinates and lattice matrices. The resulting configurations define a fixed Oct–Sqr structural path on which the same set of geometries can be evaluated using different electronic-structure treatments. Because the path endpoints were uniformly relaxed using non-spin-polarized PBE rather than separately at each electronic-structure level, this construction should be interpreted as a demonstrative comparison of reference-level energetic differences, rather than as a minimum-energy transition path or a determination of equilibrium phase stability.

Detailed PBE, PBE+U, r2SCAN, and HSE06 results, together with magnetic-moment analysis, are provided in Supplementary Section 3. Along this fixed path, non-spin-polarized PBE and FM-initialized PBE+U exhibit strongly opposing Oct–Sqr energetic preferences, providing a stringent testbed for evaluating whether an MLFF previously adapted to one reference surface can subsequently be redirected to the other. Additional spin-polarized PBE control calculations show that spin polarization alone does not reproduce this energetic reversal; the PBE Sqr solution is metallic and essentially nonmagnetic, whereas PBE+U supports distinct metallic HS-like and semiconducting LS-like solutions, as detailed in Supplementary Sections 3.2 and 3.3. These comparisons also suggest that, although r2SCAN-based pretraining has been recommended in several recent studies^[31-33] as a route toward improved thermodynamic accuracy and zero-shot phase predictions, using a higher-level semilocal reference such as r2SCAN may not by itself eliminate the need for system-specific validation and, where necessary, targeted fine-tuning.

We next examine whether pretrained MLFFs can be efficiently adapted to the FM-initialized PBE+U reference surface. Figure 2 shows the energy, force, and virial RMSEs on the held-out PBE+U test set as functions of the number of target-protocol training configurations, N_train. The results at N_train = 0 correspond to zero-shot predictions. We compare four pretrained models spanning three architectures: DPA3, DPA4-Air, and DPA4-Neo. For DPA3, we consider two pretrained branches, DPA3-MP and DPA3-OMat, trained on the MP^[7] and OMat24^[11] datasets, respectively. For DPA4-Air and DPA4-Neo, we use the separately released DPA4-Air-MP and DPA4-Neo-OMat checkpoints, pretrained on MP and OMat24, respectively. For each N_train > 0, we train and evaluate each model on the three predefined data splits; within each split, the test set remains fixed across all training-set sizes. For the zero-shot case, we evaluate the pretrained models on the corresponding test sets.

All four models improve substantially upon fine-tuning with a small amount of PBE+U data. In particular, the test-set energy-offset-aligned zero-shot RMSEs of the DPA-4 models are approximately 15-20 meV/atom and decrease to approximately 1-2 meV/atom with only 50 training configurations and approach 0.5 meV/atom after fine-tuning on 170 configurations. Their force RMSEs similarly decrease from approximately 125-175 meV/Å for zero-shot predictions to approximately 30 meV/Å with 170 training configurations and improve further with additional training data. Their virial RMSEs decrease from approximately 50-80 meV/atom to around 5 meV/atom. Exact split-wise RMSEs at N_train = 170, together with three-split summary statistics, are provided in Supplementary Section 5.1. Compared with training DPA4-Neo from scratch, fine-tuning from the pretrained DPA4-Neo-OMat initialization consistently yields lower force and virial RMSEs, particularly in the low-data regime. These results show that foundation MLFFs pretrained on large materials datasets can be transferred efficiently to a consistently defined, system-specific PBE+U surface, despite differences between the electronic-structure conventions represented during pretraining and the consistently defined target level used for fine-tuning.

Under the present fine-tuning protocol (see Section “Foundation MLFFs and fine-tuning”), the tested DPA-4 models exhibit greater data efficiency than the tested DPA-3 models. Although the DPA-3 models also benefit from target-protocol fine-tuning, their energy RMSEs remain approximately 1-2 meV/atom over much of the investigated range, and their force RMSEs reach approximately 30 meV/Å only for the largest training subsets (N_train = 290), while their virial RMSEs also remain substantially higher. In comparison, both DPA-4 models reach lower energy, force, and virial RMSEs with fewer PBE+U labels.

We further test a more demanding transfer scenario in which the pretrained models are first fine-tuned on non-spin-polarized PBE data without U and subsequently transferred to the FM-initialized PBE+U target surface. The intermediate fine-tuning stage deliberately fits the models to the potential-energy surface that favors Sqr over Oct, allowing us to test whether prior learning of a qualitatively distinct energetic preference impedes subsequent target-protocol adaptation.

To verify accurate fitting to the no-U reference surface, we examine DPA4-Neo-OMat, which was first fine-tuned using 660 non-spin-polarized PBE configurations. The corresponding DFT-label standard deviations are reported in the parity plots to provide the scale of variation in each target quantity. As shown in Figure 3A-C, the resulting model reproduces the test-set no-U energies, forces, and virials with RMSEs of 0.12 meV/atom, 6.49 meV/Å and 4.88 meV/atom, respectively. The intermediate model therefore reproduces the no-U surface accurately, rather than merely acquiring only a weak source-protocol bias.

Figure 3D-F show the test errors obtained when the no-U-adapted models are further fine-tuned using FM-initialized PBE+U data. Despite their prior alignment with the opposing energetic preference, both DPA-4 models adapt rapidly to the PBE+U surface. Their RMSEs decrease with increasing training-set size at rates comparable to those obtained by direct fine-tuning from the corresponding foundation models in Figure 2. For the representative split 0 at N_train = 170, the sequentially adapted DPA4-Neo reaches energy, force, and virial RMSEs of 0.59 meV/atom, 31.96 meV/Å, and 6.13 meV/atom, respectively, as further shown in Figure 3G-I. Statistics across all three splits are provided in Supplementary Section 5.1. The Sqr configurations exhibit consistently larger errors than the Oct configurations across 300, 600, and 900 K despite comprising approximately two-thirds of the N_train = 170 training subsets; detailed phase compositions and phase-resolved RMSEs are provided in Supplementary Section 5.2. Therefore, prior alignment with the no-U surface does not substantially increase the number of target-protocol configurations required by DPA-4. We further evaluated the directly and sequentially adapted DPA4-Neo models on independently initialized 600 K PBE+U AIMD trajectories that were not used during fine-tuning [Supplementary Figure 3]. The resulting errors are higher than those obtained from the original trajectory-based test splits, with total energy, force, and virial RMSEs of 3.15 meV/atom, 61.62 meV/Å, and 11.16 meV/atom for direct fine-tuning, and 3.73 meV/atom, 65.09 meV/Å, and 11.02 meV/atom for sequential adaptation. Because 600 K is represented in the training data and does not show systematically larger errors in the original held-out test sets, the increase on the independent trajectories cannot be attributed primarily to a temperature-distribution shift. Instead, the degradation is localized mainly to Sqr configurations and is consistent with a configurational/electronic-state distribution shift in the multistable Sqr region [Supplementary Sections 1.3 and 5.3]. Nevertheless, both adapted models benefit substantially from fine-tuning compared with the zero-shot model, and the sequentially adapted model recovers nearly the same accuracy as direct fine-tuning, providing further evidence that transfer between the two electronic-structure levels remains effective beyond configurations directly correlated with the original AIMD dataset.

More importantly, second-stage fine-tuning corrects not only the aggregate test errors but also the target-protocol energetic trend. Figure 4 compares the relative-energy profiles predicted by the no-U-adapted, directly PBE+U-fine-tuned, and sequentially adapted DPA4-Neo models along the common Oct–Sqr interpolation path. The no-U-adapted model places the Sqr endpoint 0.796 eV/f.u. below Oct. In contrast, after second-stage fine-tuning with PBE+U, the recovered model places Sqr 0.765 eV/f.u. above Oct, reproducing the sign and closely following the profile obtained from direct PBE+U fine-tuning. The inherited source-protocol energetic preference can therefore be reversed, and a compact target-protocol dataset is sufficient to redirect DPA-4 toward the opposing energetic preference. Although both the direct and sequential PBE+U fine-tuning routes yield a positive Sqr–Oct energy difference, both underestimate its magnitude, predicting 0.765 eV/f.u. compared with the PBE+U value of 0.936 eV/f.u. Our additional analyses indicate that this residual discrepancy is localized primarily near the Sqr end of the path and is most plausibly associated with the particularly complex electronic/magnetic landscape of Sqr NiO, where closely related or even identical atomic geometries can converge to distinct SCF states with substantially different energies. This multistability persists when the electronic SCF convergence threshold is tightened from 10^-6 to 10^-8 eV, indicating that it is not simply a consequence of insufficient SCF convergence [Supplementary Section 1.3]. Detailed analyses of the structural coverage, electronic/magnetic multistability, and its consequences for a coordinate-only MLFF are provided in Supplementary Section 1.

DPA-3 also benefits from second-stage fine-tuning, but its energy, force, and virial RMSEs remain substantially higher than those of the DPA-4 models over the investigated training-set sizes, with greater variation across the predefined data splits. DPA-4 therefore exhibits greater adaptability and greater stability across the tested data splits. The contrast in recovery behavior of DPA-3 and DPA-4 further indicates that efficient cross-protocol adaptation is not an automatic consequence of fine-tuning, but rather a model-dependent capability of the underlying MLFF.