Physics-Informed Weakly Supervised Learning For Interatomic Potentials

Dear Reviewer,

Thank you for your constructive and encouraging feedback. We have carefully considered your comments and will incorporate the necessary revisions in the camera-ready version. Below, we provide detailed responses to each of your encouraging proposals and questions. Please let us know if any of your concerns remain unaddressed.

$\**Q1**$. We agree that evaluating PIWSL on additional datasets would be beneficial. However, we have evaluated PIWSL across various MLIP models and datasets, including heterogeneous datasets such as ANI-1x. As recommended, we will discuss the limitations of the benchmark dataset in the "Limitations" section and plan to explore a broader range of datasets in future work.

$\**W2/Q2**$. While a third-order extension may improve accuracy, the computational cost of calculating second and third-order derivatives of the potential may outweigh the benefits. An alternative approach would be to explore new PISC spatial configurations, which offer greater flexibility than PITC, as discussed in Appendix E.2. We will mention the computational cost trade-off mentioned in Weakness 2 in the main-body in the camera-ready version. The performance improvements achieved by PIWSL justify the additional computational cost. For example, Table A20 compares training with extended iterations, showing that while the baseline method overfits the training data, PIWSL continues to improve. This is particularly important for the sparse data regime, which is our primary focus. Moreover, the increase in training time is negligible compared to the substantial computational cost and time required for data generation, particularly when using DFT or coupled-cluster methods. This becomes even more significant as molecular size increases.

$**Q3**$. Thank you for your insightful comment! PITC is applicable whenever the target property depends on atomic coordinates. For instance, DeNS, the recommended work by the reviewer, applied their method to relaxation energy estimation in OC20/22, and we expect that PIWSL could also be effectively applied to this work. We are excited about the potential to explore new application domains in future research.

$**Comparison with DeNS (Liao+2024)**$

We appreciate the recommendation of DeNS. We acknowledge that DeNS was recently accepted on December 27th. Although we have already cited it in Appendix C.2 as a related work, we will provide a more detailed discussion of these differences in the “Related Work” section of the main body in the camera-ready version.

The key differences between PIWSL and DeNS are as follows:

1. DeNS requires force labels, whereas PIWSL does not. This allows PIWSL to fine-tune MLIP models using datasets containing only potential energies, as demonstrated in Table 3. This scenario is particularly interesting for methods like CC/CBS.

2. While DeNS introduces a smaller increase in training time, PIWSL offers greater flexibility by eliminating the need for force supervision, making it more broadly applicable in scenarios where force labels are unavailable.

3. DeNS incorporates an equivariant force encoding module within the target MLIP model, which requires integration into the training pipeline. In contrast, PIWSL directly refines energy-based models without such modifications.

4. DeNS primarily discusses energy and force errors but does not explicitly discuss robustness in molecular dynamics (MD) simulations, which is one of the key aspects in our work.

Additionally, an interesting open question is whether techniques from DeNS could complement PIWSL to further enhance MLIP performance. While we have not explored this combination in this work, investigating such synergies could be valuable for future research.

Dear Reviewer,

Thank you for your constructive feedback. We have carefully considered your comments and will incorporate the necessary revisions in the camera-ready version. Below, we provide detailed responses to each point.

$\**W1**$: A detailed discussion regarding the computational time required for both the baseline and PIWSL is provided in Tables A10 and A11 in Appendix E2. Additionally, data sample efficiency is partly analyzed in Table 3, demonstrating that PIWSL achieves comparable or better accuracy while requiring only half the training data for finetuning a foundation model. Since our primary focus is on the sparse data regime, as explained in the paper, data efficiency is a critical factor for us. This makes PIWSL particularly valuable in our target setting. We will reference these results more prominently in the main paper.

$\**W2**$: We acknowledge the reviewer’s concern that PIWSL’s relative improvement decreases as dataset size increases. This trend is expected because, in data-rich scenarios, supervised learning already provides strong generalization, reducing the additional benefit of weak labels. However, PIWSL remains valuable, particularly in materials discovery, where data is scarce and MLIP models must be trained from scratch or fine-tuned from foundation models. Even in data-rich settings, PIWSL still provides meaningful improvements. For instance, with the ANI-1x 5M dataset, PIWSL achieves over a 10% error reduction, demonstrating that weak labels enhance learning even when abundant supervised data is available. While the marginal gains decrease, the improvements remain significant given the high precision requirements in MLIP applications.

$**Q1**$: Although training stability depends on the coefficients of PITC and PISC, we do not observe instability in the initial stages of training with PIWSL. This is because, at the beginning of training, the energy and force regression losses are significantly larger than the PIWSL losses. While increasing the PIWSL loss coefficients could lead to instability in the initial phase, it would also degrade overall performance. If instability were to arise, a potential mitigation strategy would be gradually increasing the PIWSL loss weight during training (similar to curriculum learning).

$**Q2**$: As stated in the final sentence of Section 5.2 ("Heterogeneous Molecular Dataset"), we attribute this issue to NoisyNode's inability to properly capture the response of energy and atomic forces to perturbations in atomic positions.

Dear Reviewer,

Thank you for your constructive feedback. We have carefully considered your comments and will incorporate the necessary revisions in the camera-ready version. Below, we provide detailed responses to each point.

$\**Q1**$. The iterations in Table A2 apply to both the baseline and PIWSL. Baseline methods were trained until convergence based on standard criteria, ensuring a fair comparison. Table A20 shows that while the baseline overfits with prolonged training, PIWSL continues improving due to physics-informed weak labels, which is especially beneficial in sparse data regimes.

To further support this, we extended MACE-OFF training:

\\begin{array} {|r|r|}\\hline & Model & Epoch & E-RMSE & F-RMSE \\\ \\hline MACE-OFF & Baseline & 100 & 1.21 & 6.90 \\\ \\hline & & 1000 & 1.19 & 6.24 \\\ \\hline \\end{array}

These results confirm PIWSL’s effectiveness even when the baseline trains significantly longer. We will include this in the final version.

$**Q2**$. Below, we present new results on $\\epsilon$ dependence for fine-tuning MACE-OFF, supplementing Table 3:

\\begin{array} {|r|r|}\\hline & Model & \\epsilon & E-RMSE & F-RMSE \\\ \\hline MACE-OFF & PIWSL & 0.005 & 0.73 & 4.20 \\\ \\hline & & 0.01 (Table 3) & 0.72 & 3.77 \\\ \\hline & & 0.02 & 0.75 & 4.04 \\\ \\hline \\end{array}

These results show that PIWSL maintains stable performance across reasonable $\\epsilon$ variations. The choice of $\\epsilon$ = 0.3 of the original bond length was empirically determined to ensure the validity of the Taylor expansion. Table A18 shows that perturbations closer this threshold degrade performance, indicating the breakdown of the approximation. This aligns with constraints observed in prior studies, which we will clarify in the final version.

$**Q3**$. No. MACE does not include a direct force module; we used official MACE models. We will explicitly mention this.

$**Q4**$. The RMSE ratio is computed as RMSE_PIWSL / RMSE_Baseline to determine whether the PIWSL error is lower than that of the baseline. Thank you for your observation—we will add an explicit definition in the final version.

$**Q5**$. To clarify, CCSD/CBS was not used to generate the energy and force labels in this experiment. Instead, CCSD/cc-pVDZ was used [1]. During training, we relied solely on energy labels, while CCSD/cc-pVDZ force labels were used only in the test dataset to evaluate whether PIWSL could improve force predictions without force labels during training. This setup is motivated by real-world applications where force labels are often unavailable due to the high computational cost of advanced coupled cluster methods, such as CC/CBS. By testing in this way, we assess PIWSL’s ability to generalize force predictions from energy supervision alone.

[1] Chmiela. et al. Nat Commun 9, 3887 (2018).

$**Comparison with DeNS (Liao+2024)**$

The key differences between PIWSL and DeNS are as follows:

1. DeNS requires force labels, whereas PIWSL does not. This allows PIWSL to fine-tune MLIP models using datasets containing only potential energies, as demonstrated in Table 3. This scenario is particularly interesting for methods like CC/CBS.

2. While DeNS introduces a smaller increase in training time, PIWSL offers greater flexibility by eliminating the need for force supervision, making it more broadly applicable in scenarios where force labels are unavailable.

3. DeNS incorporates an equivariant force encoding module within the target MLIP model, which requires integration into the training pipeline. In contrast, PIWSL directly refines energy-based models without such modifications.

4. DeNS primarily discusses energy and force errors but does not explicitly discuss robustness in molecular dynamics (MD) simulations, which is one of the key aspects in our work.

Additionally, an interesting open question is whether techniques from DeNS could complement PIWSL to further enhance MLIP performance. While we have not explored this combination in this work, investigating such synergies could be valuable for future research. We acknowledge that DeNS was recently accepted on December 27th. Although we have already cited it in Appendix C.2 as a related work, we will provide a more detailed discussion of these differences in the “Related Work” section of the main body in the camera-ready version.

$**Periodic boundary condition**$

As noted by the reviewer, atoms outside the unit cell are wrapped back into the cell when noise is applied. We will add this clarification to the camera-ready version.

$**Physics-Informed Claim for PITC**$

Concerning the physical motivation behind PITC: It is valid to locally approximate a potential energy surface using a Taylor series.