The ability to perform fast and accurate atomistic simulations is crucial for advancing the chemical sciences. By learning from high-quality data, machine-learned interatomic potentials achieve accuracy on par with ab initio and first-principles methods at a fraction of their computational cost. The success of machine-learned interatomic potentials arises from integrating inductive biases such as equivariance to group actions on an atomic system, e.g., equivariance to rotations and reflections. In particular, the field has notably advanced with the emergence of equivariant message passing. Most of these models represent an atomic system using spherical tensors, tensor products of which require complicated numerical coefficients and can be computationally demanding. Cartesian tensors offer a promising alternative, though state-of-the-art methods lack flexibility in message-passing mechanisms, restricting their architectures and expressive power. This work explores higher-rank irreducible Cartesian tensors to address these limitations. We integrate irreducible Cartesian tensor products into message-passing neural networks and prove the equivariance and traceless property of the resulting layers. Through empirical evaluations on various benchmark data sets, we consistently observe on-par or better performance than that of state-of-the-art spherical and Cartesian models.
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Summary
This work builds upon advances on equivariant and many-body architectures for the construction of neural network potentials. It lays out the formalism to substitute the conventionally-used spherical tensors in higher-rank models for Cartesian tensors. Taking as a reference the MACE architecture, the authors intend to show that this is a competitive approach to SOTA models in terms of accuracy and computational efficiency.
Strengths
The exploration of new methods to more efficiently learn machine learning force fields is an active area of research, and the use of higher-rank Cartesian approaches is quite novel, in contrast to the use of the spherical basis. The authors demonstrate that they can obtain results on benchmarks datasets that compete with SOTA models (both spherical and higher-rank Cartesian). The exposition of mathematical concepts is quite clear for a reader familiar with the literature. In terms of accuracy, the model is very satisfactory.
Weaknesses
Although the achievement of competitive performance when compared to SOTA models is relevant enough, and the mathematical machinery is novel for the neural network potential field, I am not sure whether the authors have been able to demonstrate in some way why their method should be chosen in contrast to MACE, for example. The comparison of inference times seems to not favor the use of ICTP. However, I am aware of the fact that the formalism laid out in the paper allows the construction of other architectures, and that the design space of these models could be further investigated to find even more efficient models.
Questions
1) It seems that the first model to explore the idea of higher-rank Cartesian tensors was TensorNet [30], even though it is not flexible to incorporate arbitrary ranks, and it does not explicitly account for many-body interactions. I miss some more thorough discussion on their differences, even more taking into account that [30] seems to display competitive performance to ICTP without making use of those more sophisticated approaches. I would encourage the authors to include some discussion in this regard. Is ICTP a combination of CACE and TensorNet? 2) I do not intend the authors to address the following question with more experiments, I acknowledge the limited time frame, but: experiments have been conducted on datasets consisting of single systems. Do the authors have any reference of how the model performs on datasets with varying chemical composition?
Rating
7
Confidence
3
Soundness
3
Presentation
3
Contribution
2
Limitations
The authors have adequately addressed the limitations
Tab. 1: Inference times and memory consumption as a function of L and ν for the 3BPA data set.
All values are obtained by averaging over five independent runs. Best performances are highlighted in bold. Inference time and memory consumption are measured for a batch size of 10. Inference time is reported per structure in ms; memory consumption is provided for the entire batch in GB. | | *L* = 1 | | *L* = 2 | | *L* = 3 | | |:---|:---|:---|:---|:---|:---|:---| | | ICTP | MACE | ICTP | MACE | ICTP | MACE | | Inference times | | | | | | | | *ν* = 1 | **0.76 ± 0.17** | 1.02 ± 0.03 | **0.87 ± 0.18** | 1.38 ± 0.04 | **0.98 ± 0.26** | 1.88 ± 0.03 | | *ν* = 2 | **0.59 ± 0.20** | 1.12 ± 0.03 | **1.03 ± 0.21** | 1.52 ± 0.05 | **1.34 ± 0.08** | 2.0 ± 0.10 | | *ν* = 3 | **0.79 ± 0.22** | 1.23 ± 0.03 | **1.15 ± 0.08** | 1.67 ± 0.03 | **1.85 ± 0.13** | 2.23 ± 0.03 | | *ν* = 4 | **0.94 ± 0.17** | 1.41 ± 0.11 | **1.31 ± 0.21** | 1.83 ± 0.01 | **2.07 ± 0.20** | 2.53 ± 0.01 | | *ν* = 5 | **1.02 ± 0.17** | 1.52 ± 0.08 | **1.72 ± 0.07** | 2.26 ± 0.03 | **3.61 ± 0.02** | OOM | | *ν* = 6 | **1.00 ± 0.07** | 1.77 ± 0.05 | **1.83 ± 0.16** | 27.85 ± 0.01 | **16.76 ± 0.35** | OOM | | Memory consumption | | | | | | | | *ν* = 1 | 0.05 ± 0.00 | **0.04 ± 0.00** | 0.08 ± 0.00 | **0.06 ± 0.00** | 0.21 ± 0.00 | **0.13 ± 0.00** | | *ν* = 2 | 0.05 ± 0.00 | **0.04 ± 0.00** | 0.08 ± 0.00 | **0.07 ± 0.00** | 0.28 ± 0.09 | **0.13 ± 0.00** | | *ν* = 3 | 0.05 ± 0.00 | **0.04 ± 0.00** | 0.10 ± 0.00 | **0.08 ± 0.00** | 0.51 ± 0.03 | **0.23 ± 0.00** | | *ν* = 4 | **0.05 ± 0.00** | **0.05 ± 0.00** | **0.18 ± 0.08** | 0.30 ± 0.00 | **1.07 ± 0.10** | 4.16 ± 0.00 | | *ν* = 5 | **0.05 ± 0.00** | 0.07 ± 0.00 | **0.35 ± 0.07** | 3.18 ± 0.00 | **5.07 ± 0.02** | OOM | | *ν* = 6 | **0.11 ± 0.09** | 0.22 ± 0.00 | **0.93 ± 0.00** | 50.49 ± 0.00 | **28.48 ± 0.03** | OOM |
Tab. 2: Energy (E, meV) and force (F, meV/Å) RMSEs for the 3BPA data set and ν = 1.
All values are obtained by averaging over five independent runs. Best performances are highlighted in bold. Inference time and memory consumption are measured for a batch size of 100. Inference time is reported per structure in ms; memory consumption is provided for the entire batch in GB. | | | ICTP (*L* = 2) | MACE (*L* = 2) | |:-------------------------------|:---:|-------------------:|-----------------:| | 300 K | E | **12.90 ± 1.06** | **13.50 ± 1.71** | | | F | **29.90 ± 0.25** | **30.18 ± 0.38** | | 600 K | E | **29.97 ± 0.94** | **31.32 ± 2.16** | | | F | **62.80 ± 0.45** | **63.04 ± 0.73** | | 1200 K | E | **81.03 ± 1.64** | **81.54 ± 2.02** | | | F | **146.96 ± 1.30** | 149.44 ± 1.94 | | Dihedral slices | E | **22.84 ± 2.96** | 28.08 ± 4.04 | | | F | **48.82 ± 5.25** | **49.62 ± 2.92** | | Inference time | | **2.62 ± 0.02** | 2.96 ± 0.06 | | Memory consumption | | 32.57 ± 0.00 | **23.32 ± 0.00** |
Tab. 3: Energy (E, meV) and force (F, eV/Å) RMSEs for Ta-V-Cr-W subsystems.
Results are obtained by averaging over 10 independent runs. Best performances are highlighted in bold. Inference time and memory consumption are measured for a batch size of 50. Inference time is reported per atom in μs; memory consumption is provided for the entire batch in GB. | Subsystem | | ICTP (*L* = 2) | ICTP (*L* = 1) | ICTP (*L* = 0) | MACE (*L* = 2) | MACE (*L* = 1) | MACE (*L* = 0) | ICTP (*L* = 2, *ν* = 2) | MTP | GM-NN | EAM | |:---|:--:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:| | TaV | E | **1.02 ± 0.27** | **1.21 ± 0.54** | 1.65 ± 1.06 | 1.72 ± 0.67 | 1.76 ± 0.53 | 2.24 ± 1.34 | **1.24 ± 0.50** | 1.94 | 1.54 | 32.0 | | | F | **0.020 ± 0.002** | 0.022 ± 0.002 | 0.024 ± 0.002 | **0.019 ± 0.002** | **0.020 ± 0.003** | 0.022 ± 0.002 | 0.023 ± 0.002 | 0.050 | 0.029 | 0.404 | | TaCr | E | **1.81 ± 0.29** | **1.94 ± 0.23** | 2.13 ± 0.19 | 3.26 ± 0.42 | 3.31 ± 0.44 | 4.18 ± 0.56 | 2.4 ± 0.33 | 3.26 | 2.98 | 43.6 | | | F | **0.025 ± 0.007** | **0.024 ± 0.006** | 0.027 ± 0.005 | 0.029 ± 0.01 | **0.026 ± 0.007** | 0.028 ± 0.007 | **0.026 ± 0.006** | 0.057 | 0.038 | 0.343 | | TaW | E | **1.75 ± 0.11** | 1.87 ± 0.14 | 2.45 ± 0.31 | 2.73 ± 0.53 | 3.21 ± 0.55 | 3.57 ± 0.48 | 2.19 ± 0.54 | 2.72 | 2.99 | 44.8 | | | F | **0.017 ± 0.002** | **0.018 ± 0.002** | 0.020 ± 0.002 | **0.017 ± 0.002** | **0.018 ± 0.002** | 0.019 ± 0.002 | **0.018 ± 0.002** | 0.038 | 0.025 | 0.248 | | VCr | E | **1.74 ± 1.2** | 2.52 ± 2.43 | **2.13 ± 1.24** | **2.19 ± 0.78** | 2.82 ± 1.28 | 3.11 ± 1.42 | **1.89 ± 1.27** | **2.29** | 2.82 | 44.8 | | | F | **0.016 ± 0.002** | 0.018 ± 0.001 | 0.019 ± 0.001 | **0.016 ± 0.001** | **0.017 ± 0.001** | 0.018 ± 0.002 | 0.019 ± 0.001 | 0.036 | 0.025 | 0.270 | | VW | E | **1.32 ± 0.2** | **1.46 ± 0.16** | 1.69 ± 0.21 | 1.9 ± 0.19 | 1.94 ± 0.23 | 2.42 ± 0.24 | 1.61 ± 0.16 | 2.50 | 2.00 | 21.3 | | | F | **0.014 ± 0.002** | **0.015 ± 0.002** | 0.018 ± 0.003 | **0.014 ± 0.002** | **0.015 ± 0.002** | 0.017 ± 0.002 | 0.016 ± 0.002 | 0.037 | 0.023 | 0.292 | | CrW | E | **2.18 ± 0.93** | **2.45 ± 1.53** | 2.76 ± 1.15 | **2.31 ± 1.18** | 2.84 ± 0.98 | 4.14 ± 1.38 | 3.12 ± 1.90 | 4.35 | 2.87 | 23.4 | | | F | **0.018 ± 0.004** | **0.020 ± 0.005** | 0.024 ± 0.008 | **0.020 ± 0.009** | **0.019 ± 0.006** | 0.023 ± 0.007 | 0.022 ± 0.006 | 0.041 | 0.029 | 0.248 | | TaVCr | E | **0.79 ± 0.08** | 0.92 ± 0.17 | 1.00 ± 0.24 | 2.26 ± 0.54 | 2.71 ± 0.66 | 3.92 ± 0.77 | 0.97 ± 0.13 | 2.43 | 1.97 | 34.1 | | | F | 0.027 ± 0.001 | 0.029 ± 0.002 | 0.033 ± 0.002 | **0.023 ± 0.002** | **0.024 ± 0.001** | 0.028 ± 0.001 | 0.031 ± 0.002 | 0.054 | 0.045 | 0.313 | | TaVW | E | **1.00 ± 0.2** | **0.98 ± 0.18** | 1.26 ± 0.23 | 1.8 ± 0.35 | 1.97 ± 0.44 | 2.29 ± 0.86 | **0.95 ± 0.25** | 1.67 | 1.70 | 39.6 | | | F | **0.021 ± 0.001** | 0.022 ± 0.001 | 0.025 ± 0.001 | **0.021 ± 0.002** | 0.023 ± 0.001 | 0.026 ± 0.001 | 0.023 ± 0.001 | 0.043 | 0.034 | 0.321 | | TaCrW | E | **1.16 ± 0.15** | **1.28 ± 0.13** | 1.58 ± 0.29 | 1.67 ± 0.38 | 1.48 ± 0.50 | 2.08 ± 0.57 | **1.24 ± 0.11** | 2.08 | 2.19 | 23.6 | | | F | **0.022 ± 0.001** | 0.024 ± 0.001 | 0.027 ± 0.001 | 0.028 ± 0.002 | 0.030 ± 0.002 | 0.033 ± 0.002 | 0.026 ± 0.001 | 0.051 | 0.039 | 0.327 | | VCrW | E | **1.00 ± 0.16** | **1.07 ± 0.14** | 1.37 ± 0.13 | 1.97 ± 0.5 | 2.21 ± 0.42 | 2.86 ± 0.64 | **1.10 ± 0.14** | 1.37 | 1.94 | 19.4 | | | F | **0.018 ± 0.001** | 0.019 ± 0.001 | 0.022 ± 0.001 | **0.017 ± 0.001** | 0.019 ± 0.001 | 0.021 ± 0.001 | 0.020 ± 0.001 | 0.040 | 0.031 | 0.314 | | TaVCrW (0 K) | E | **1.22 ± 0.07** | 1.30 ± 0.1 | 1.48 ± 0.16 | 2.26 ± 0.55 | 2.48 ± 0.46 | 3.60 ± 0.54 | **1.33 ± 0.17** | 2.09 | 2.16 | 50.8 | | | F | **0.021 ± 0.002** | **0.022 ± 0.002** | 0.025 ± 0.002 | **0.022 ± 0.001** | 0.023 ± 0.002 | 0.027 ± 0.001 | 0.024 ± 0.002 | 0.049 | 0.037 | 0.488 | | TaVCrW (2500 K) | E | **1.63 ± 0.07** | 1.74 ± 0.11 | 2.09 ± 0.09 | 2.22 ± 0.48 | 2.34 ± 0.59 | 3.68 ± 0.70 | 2.06 ± 0.09 | 2.40 | 2.67 | 59.4 | | | F | **0.116 ± 0.002** | 0.121 ± 0.002 | 0.141 ± 0.003 | **0.119 ± 0.007** | 0.126 ± 0.006 | 0.150 ± 0.003 | 0.140 ± 0.002 | 0.156 | 0.179 | 0.521 | | Overall | E | **1.38 ± 0.09** | 1.56 ± 0.21 | 1.80 ± 0.18 | 2.19 ± 0.31 | 2.42 ± 0.31 | 3.17 ± 0.28 | 1.67 ± 0.21 | 2.43 | 2.32 | 37.14 | | | F | **0.028 ± 0.001** | **0.029 ± 0.001** | 0.034 ± 0.001 | **0.029 ± 0.001** | 0.030 ± 0.001 | 0.034 ± 0.001 | 0.032 ± 0.001 | 0.054 | 0.043 | 0.443 | | Inference time | | 51.78 ± 1.18 | 25.09 ± 0.02 | 14.59 ± 0.01 | 29.48 ± 0.23 | 15.37 ± 0.04 | 4.43 ± 0.00 | 14.97 ± 0.09 | 17.57 | 7.25 | 0.50 | | Memory consumption | | 36.78 ± 0.00 | 16.93 ± 0.00 | 8.48 ± 0.00 | 28.82 ± 0.00 | 13.87 ± 0.00 | 5.91 ± 0.00 | 13.15 ± 0.00 | – | – | – |
Summary
This paper introduces the use of higher-rank irreducible Cartesian tensors as an alternative to spherical tensors for equivariant message passing in machine learning interatomic potentials. The authors illustrate clearly on how to construct these tensors and their products, prove equivariance properties, and evaluate the approach empirically on several molecular datasets.
Strengths
* The mathematical foundations are clearly illustrated, with detailed explanations of how to construct irreducible Cartesian tensors and compute their products. * The experiments on out-of-domain extrapolation, particularly on the 3BPA and acetylacetone datasets, provide valuable insights into the generalization capabilities of the proposed method. * The paper demonstrates that irreducible Cartesian tensor-based models can achieve comparable or sometimes better performance than state-of-the-art spherical tensor models.
Weaknesses
* The empirical evaluation is limited to relatively simple molecular datasets. The paper would be strengthened by including experiments on more challenging datasets such as MD22 or heterogeneous datasets like QM9. * The efficiency gain and the performance gain is not that appealing to my eye. It seems little more than “instead of using that math, you can use this math!” without a very strong theoretical justification for why to do so. The author could have done a better job of explaining what is the fundamental difference/advantage of the proposed cartesian tensors when compared with the sphereical tensors.
Questions
* What are the core differences between this method and TensorNet? A clearer comparison would help position this work in the context of existing literature. * Is the proposed model compatible with Hamiltonian prediction? This could be an interesting avenue for future work. * Can the authors provide plots showing how their model scales with increasing L? * The paper mentions "transferability" in line 225. Could the authors clarify what they mean by this term in this context? * The claim that Cartesian tensors are advantageous to spherical tensors requires further explanation. From a representation power perspective, aren't they equivalent? Is it possible that the observed performance gains are due to hyperparameter tuning rather than fundamental differences in representation power?
Rating
5
Confidence
4
Soundness
2
Presentation
3
Contribution
2
Limitations
See above.
Tab. 1: Inference times and memory consumption as a function of L and ν for the 3BPA data set.
All values are obtained by averaging over five independent runs. Best performances are highlighted in bold. Inference time and memory consumption are measured for a batch size of 10. Inference time is reported per structure in ms; memory consumption is provided for the entire batch in GB. | | *L* = 1 | | *L* = 2 | | *L* = 3 | | |:---|:---|:---|:---|:---|:---|:---| | | ICTP | MACE | ICTP | MACE | ICTP | MACE | | Inference times | | | | | | | | *ν* = 1 | **0.76 ± 0.17** | 1.02 ± 0.03 | **0.87 ± 0.18** | 1.38 ± 0.04 | **0.98 ± 0.26** | 1.88 ± 0.03 | | *ν* = 2 | **0.59 ± 0.20** | 1.12 ± 0.03 | **1.03 ± 0.21** | 1.52 ± 0.05 | **1.34 ± 0.08** | 2.0 ± 0.10 | | *ν* = 3 | **0.79 ± 0.22** | 1.23 ± 0.03 | **1.15 ± 0.08** | 1.67 ± 0.03 | **1.85 ± 0.13** | 2.23 ± 0.03 | | *ν* = 4 | **0.94 ± 0.17** | 1.41 ± 0.11 | **1.31 ± 0.21** | 1.83 ± 0.01 | **2.07 ± 0.20** | 2.53 ± 0.01 | | *ν* = 5 | **1.02 ± 0.17** | 1.52 ± 0.08 | **1.72 ± 0.07** | 2.26 ± 0.03 | **3.61 ± 0.02** | OOM | | *ν* = 6 | **1.00 ± 0.07** | 1.77 ± 0.05 | **1.83 ± 0.16** | 27.85 ± 0.01 | **16.76 ± 0.35** | OOM | | Memory consumption | | | | | | | | *ν* = 1 | 0.05 ± 0.00 | **0.04 ± 0.00** | 0.08 ± 0.00 | **0.06 ± 0.00** | 0.21 ± 0.00 | **0.13 ± 0.00** | | *ν* = 2 | 0.05 ± 0.00 | **0.04 ± 0.00** | 0.08 ± 0.00 | **0.07 ± 0.00** | 0.28 ± 0.09 | **0.13 ± 0.00** | | *ν* = 3 | 0.05 ± 0.00 | **0.04 ± 0.00** | 0.10 ± 0.00 | **0.08 ± 0.00** | 0.51 ± 0.03 | **0.23 ± 0.00** | | *ν* = 4 | **0.05 ± 0.00** | **0.05 ± 0.00** | **0.18 ± 0.08** | 0.30 ± 0.00 | **1.07 ± 0.10** | 4.16 ± 0.00 | | *ν* = 5 | **0.05 ± 0.00** | 0.07 ± 0.00 | **0.35 ± 0.07** | 3.18 ± 0.00 | **5.07 ± 0.02** | OOM | | *ν* = 6 | **0.11 ± 0.09** | 0.22 ± 0.00 | **0.93 ± 0.00** | 50.49 ± 0.00 | **28.48 ± 0.03** | OOM |
Tab. 2: Energy (E, meV) and force (F, meV/Å) RMSEs for the 3BPA data set and ν = 1.
All values are obtained by averaging over five independent runs. Best performances are highlighted in bold. Inference time and memory consumption are measured for a batch size of 100. Inference time is reported per structure in ms; memory consumption is provided for the entire batch in GB. | | | ICTP (*L* = 2) | MACE (*L* = 2) | |:-------------------------------|:---:|-------------------:|-----------------:| | 300 K | E | **12.90 ± 1.06** | **13.50 ± 1.71** | | | F | **29.90 ± 0.25** | **30.18 ± 0.38** | | 600 K | E | **29.97 ± 0.94** | **31.32 ± 2.16** | | | F | **62.80 ± 0.45** | **63.04 ± 0.73** | | 1200 K | E | **81.03 ± 1.64** | **81.54 ± 2.02** | | | F | **146.96 ± 1.30** | 149.44 ± 1.94 | | Dihedral slices | E | **22.84 ± 2.96** | 28.08 ± 4.04 | | | F | **48.82 ± 5.25** | **49.62 ± 2.92** | | Inference time | | **2.62 ± 0.02** | 2.96 ± 0.06 | | Memory consumption | | 32.57 ± 0.00 | **23.32 ± 0.00** |
Tab. 3: Energy (E, meV) and force (F, eV/Å) RMSEs for Ta-V-Cr-W subsystems.
Results are obtained by averaging over 10 independent runs. Best performances are highlighted in bold. Inference time and memory consumption are measured for a batch size of 50. Inference time is reported per atom in μs; memory consumption is provided for the entire batch in GB. | Subsystem | | ICTP (*L* = 2) | ICTP (*L* = 1) | ICTP (*L* = 0) | MACE (*L* = 2) | MACE (*L* = 1) | MACE (*L* = 0) | ICTP (*L* = 2, *ν* = 2) | MTP | GM-NN | EAM | |:---|:--:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:| | TaV | E | **1.02 ± 0.27** | **1.21 ± 0.54** | 1.65 ± 1.06 | 1.72 ± 0.67 | 1.76 ± 0.53 | 2.24 ± 1.34 | **1.24 ± 0.50** | 1.94 | 1.54 | 32.0 | | | F | **0.020 ± 0.002** | 0.022 ± 0.002 | 0.024 ± 0.002 | **0.019 ± 0.002** | **0.020 ± 0.003** | 0.022 ± 0.002 | 0.023 ± 0.002 | 0.050 | 0.029 | 0.404 | | TaCr | E | **1.81 ± 0.29** | **1.94 ± 0.23** | 2.13 ± 0.19 | 3.26 ± 0.42 | 3.31 ± 0.44 | 4.18 ± 0.56 | 2.4 ± 0.33 | 3.26 | 2.98 | 43.6 | | | F | **0.025 ± 0.007** | **0.024 ± 0.006** | 0.027 ± 0.005 | 0.029 ± 0.01 | **0.026 ± 0.007** | 0.028 ± 0.007 | **0.026 ± 0.006** | 0.057 | 0.038 | 0.343 | | TaW | E | **1.75 ± 0.11** | 1.87 ± 0.14 | 2.45 ± 0.31 | 2.73 ± 0.53 | 3.21 ± 0.55 | 3.57 ± 0.48 | 2.19 ± 0.54 | 2.72 | 2.99 | 44.8 | | | F | **0.017 ± 0.002** | **0.018 ± 0.002** | 0.020 ± 0.002 | **0.017 ± 0.002** | **0.018 ± 0.002** | 0.019 ± 0.002 | **0.018 ± 0.002** | 0.038 | 0.025 | 0.248 | | VCr | E | **1.74 ± 1.2** | 2.52 ± 2.43 | **2.13 ± 1.24** | **2.19 ± 0.78** | 2.82 ± 1.28 | 3.11 ± 1.42 | **1.89 ± 1.27** | **2.29** | 2.82 | 44.8 | | | F | **0.016 ± 0.002** | 0.018 ± 0.001 | 0.019 ± 0.001 | **0.016 ± 0.001** | **0.017 ± 0.001** | 0.018 ± 0.002 | 0.019 ± 0.001 | 0.036 | 0.025 | 0.270 | | VW | E | **1.32 ± 0.2** | **1.46 ± 0.16** | 1.69 ± 0.21 | 1.9 ± 0.19 | 1.94 ± 0.23 | 2.42 ± 0.24 | 1.61 ± 0.16 | 2.50 | 2.00 | 21.3 | | | F | **0.014 ± 0.002** | **0.015 ± 0.002** | 0.018 ± 0.003 | **0.014 ± 0.002** | **0.015 ± 0.002** | 0.017 ± 0.002 | 0.016 ± 0.002 | 0.037 | 0.023 | 0.292 | | CrW | E | **2.18 ± 0.93** | **2.45 ± 1.53** | 2.76 ± 1.15 | **2.31 ± 1.18** | 2.84 ± 0.98 | 4.14 ± 1.38 | 3.12 ± 1.90 | 4.35 | 2.87 | 23.4 | | | F | **0.018 ± 0.004** | **0.020 ± 0.005** | 0.024 ± 0.008 | **0.020 ± 0.009** | **0.019 ± 0.006** | 0.023 ± 0.007 | 0.022 ± 0.006 | 0.041 | 0.029 | 0.248 | | TaVCr | E | **0.79 ± 0.08** | 0.92 ± 0.17 | 1.00 ± 0.24 | 2.26 ± 0.54 | 2.71 ± 0.66 | 3.92 ± 0.77 | 0.97 ± 0.13 | 2.43 | 1.97 | 34.1 | | | F | 0.027 ± 0.001 | 0.029 ± 0.002 | 0.033 ± 0.002 | **0.023 ± 0.002** | **0.024 ± 0.001** | 0.028 ± 0.001 | 0.031 ± 0.002 | 0.054 | 0.045 | 0.313 | | TaVW | E | **1.00 ± 0.2** | **0.98 ± 0.18** | 1.26 ± 0.23 | 1.8 ± 0.35 | 1.97 ± 0.44 | 2.29 ± 0.86 | **0.95 ± 0.25** | 1.67 | 1.70 | 39.6 | | | F | **0.021 ± 0.001** | 0.022 ± 0.001 | 0.025 ± 0.001 | **0.021 ± 0.002** | 0.023 ± 0.001 | 0.026 ± 0.001 | 0.023 ± 0.001 | 0.043 | 0.034 | 0.321 | | TaCrW | E | **1.16 ± 0.15** | **1.28 ± 0.13** | 1.58 ± 0.29 | 1.67 ± 0.38 | 1.48 ± 0.50 | 2.08 ± 0.57 | **1.24 ± 0.11** | 2.08 | 2.19 | 23.6 | | | F | **0.022 ± 0.001** | 0.024 ± 0.001 | 0.027 ± 0.001 | 0.028 ± 0.002 | 0.030 ± 0.002 | 0.033 ± 0.002 | 0.026 ± 0.001 | 0.051 | 0.039 | 0.327 | | VCrW | E | **1.00 ± 0.16** | **1.07 ± 0.14** | 1.37 ± 0.13 | 1.97 ± 0.5 | 2.21 ± 0.42 | 2.86 ± 0.64 | **1.10 ± 0.14** | 1.37 | 1.94 | 19.4 | | | F | **0.018 ± 0.001** | 0.019 ± 0.001 | 0.022 ± 0.001 | **0.017 ± 0.001** | 0.019 ± 0.001 | 0.021 ± 0.001 | 0.020 ± 0.001 | 0.040 | 0.031 | 0.314 | | TaVCrW (0 K) | E | **1.22 ± 0.07** | 1.30 ± 0.1 | 1.48 ± 0.16 | 2.26 ± 0.55 | 2.48 ± 0.46 | 3.60 ± 0.54 | **1.33 ± 0.17** | 2.09 | 2.16 | 50.8 | | | F | **0.021 ± 0.002** | **0.022 ± 0.002** | 0.025 ± 0.002 | **0.022 ± 0.001** | 0.023 ± 0.002 | 0.027 ± 0.001 | 0.024 ± 0.002 | 0.049 | 0.037 | 0.488 | | TaVCrW (2500 K) | E | **1.63 ± 0.07** | 1.74 ± 0.11 | 2.09 ± 0.09 | 2.22 ± 0.48 | 2.34 ± 0.59 | 3.68 ± 0.70 | 2.06 ± 0.09 | 2.40 | 2.67 | 59.4 | | | F | **0.116 ± 0.002** | 0.121 ± 0.002 | 0.141 ± 0.003 | **0.119 ± 0.007** | 0.126 ± 0.006 | 0.150 ± 0.003 | 0.140 ± 0.002 | 0.156 | 0.179 | 0.521 | | Overall | E | **1.38 ± 0.09** | 1.56 ± 0.21 | 1.80 ± 0.18 | 2.19 ± 0.31 | 2.42 ± 0.31 | 3.17 ± 0.28 | 1.67 ± 0.21 | 2.43 | 2.32 | 37.14 | | | F | **0.028 ± 0.001** | **0.029 ± 0.001** | 0.034 ± 0.001 | **0.029 ± 0.001** | 0.030 ± 0.001 | 0.034 ± 0.001 | 0.032 ± 0.001 | 0.054 | 0.043 | 0.443 | | Inference time | | 51.78 ± 1.18 | 25.09 ± 0.02 | 14.59 ± 0.01 | 29.48 ± 0.23 | 15.37 ± 0.04 | 4.43 ± 0.00 | 14.97 ± 0.09 | 17.57 | 7.25 | 0.50 | | Memory consumption | | 36.78 ± 0.00 | 16.93 ± 0.00 | 8.48 ± 0.00 | 28.82 ± 0.00 | 13.87 ± 0.00 | 5.91 ± 0.00 | 13.15 ± 0.00 | – | – | – |
Questions on Ta-V-Cr-W system
Thank you for your response. I am curious on how you trained the Ta-V-Cr-W systems. Did you train them jointly or separately? If separately, can model train on low temperature extrapolates to high temperature? How long does it take to train the system? Also, how do you handle the heterogeneity in the system for predicting the energies and forces? Can I find a reference for the dataset?
Dear Reviewer, We thank you for your prompt response. We noticed that other reviewers cannot read your comment. Therefore, we will add it to allow them to follow our discussion: > Thank you for your response. > > I am curious on how you trained the Ta-V-Cr-W systems. Did you train them jointly or separately? If separately, can model train > on low temperature extrapolates to high temperature? How long does it take to train the system? Also, how do you handle the > heterogeneity in the system for predicting the energies and forces? Can I find a reference for the dataset? We have addressed each of your questions below: * We train ICTP and MACE using all Ta-V-Cr-W subsystems simultaneously. Particularly, as already stated in the general response, all models are trained using 5373 configurations (4873 are used for training and 500—for early stopping), while the remaining 1338 configurations are reserved for testing the models' performance. The performance is tested separately using 0 K binaries, ternaries, quaternaries, and near-melting temperature four-component disordered alloys. * Models trained exclusively on 0 K subsystems are not expected to generalize to near-melting temperature four-component disordered alloys. The 0 K subsystems span: (i) different atomic combinations for relaxed binary, ternary, and quaternary systems; (ii) different low-temperature ordering in the Ta-V-Cr-W family (B2 ordering, B32 ordering, random binary solid solution, BCC interface); (iii) all possible phase separations on the TaVCrW lattice (B2/B2 ordering, B2/B32 ordering, B32/B32 ordering, B2/random binary ordering, B32/random binary ordering, random binary/random binary ordering). None overlaps sufficiently in local environments with high-temperature (2500 K) disordered structures. For more details on the data set, we refer to the "Description of the data set" section of the original publication [C]. * Training a single model requires up to 12 hours on an NVIDIA A100 GPU with 80GB. * We did not implement any specific step for handling the heterogeneity in the Ta-V-Cr-W data set. We only increased the mini-batch size to 32 for both models to account for energy statistics and reduced the relative weight of the force loss. * For the dataset, we have referenced [C] in our previous response (K. Gubaev, V. Zaverkin, P. Srinivasan *et al.*: Performance of two complementary machine-learned potentials in modelling chemically complex systems. *npj Comput. Mater.* **9**, 129 (2023)). The data set can be accessed via the link: [https://doi.org/10.18419/darus-3516](https://doi.org/10.18419/darus-3516). We hope we could properly address your questions and awaiting on your response.
Thanks for the response
Sorry for the oversight. I was not paying so much attention to the general response. That addresses most of my concern. But I am still worried that the performance/efficiency improvements is not significant and stronger baselines (such as Equiformer V2) or more standardized datasets (such as MD22/OCP as mentioned by Reviewer 69Cm) are needed.
Dear reviewer, We again noticed that other reviewers cannot read your comment. Therefore, we will add it to allow them to follow our discussion: > Sorry for the oversight. I was not paying so much attention to the general response. That addresses most of my concern. But > I am still worried that the performance/efficiency improvements is not significant and stronger baselines (such as Equiformer > V2) or more standardized datasets (such as MD22/OCP as mentioned by Reviewer 69Cm) are needed. We want to point out that the official review does not mention comparing to an additional baseline, such as EquiformerV2. Besides that, baselines, such as MACE, Allegro, NequIP, TensorNet, and CACE, which we used in our work, are current state-of-the-art models. Therefore, we do not see how adding experimental results for EquiformerV2 could further contribute to demonstrating the performance and efficiency advantages of our approach. We evaluated ICTP using rMD17, 3BPA, and Acetylacetone, which other state-of-the-art models commonly use. Also, as requested by the reviewer, we included another, more challenging data set (Ta-V-Cr-W) and motivated our choice. Including yet another benchmark data set, such as MD22 or OC20/22, would not improve the value of our work. As we already explained, the suggested MD22 data set does not include variations in atom types. Thus, it would not provide additional insights beyond those we acquired with rMD17; the models' performance would again be tested on vibrational degrees of freedom of a single molecule. Furthermore, and contrary to the Ta-V-Cr-W data set, OC20/22 does not allow systematic evaluation of the models' performance across different crystal structures, temperatures, and stoichiometries. We would appreciate further clarification on the reviewer's concerns regarding the models and data sets used in the manuscript and the response to the official review. We are eager to better understand the specific reasons why our current evaluation is not convincing so far.
Thanks for the response
Thank you for your prompt response. I will raise my score. However, I hope that more challenging datasets will be included in the revised version. MD22 can measure long-range effects, and OCP has numerous other baseline performances reported in the literature. To provide a more comprehensive evaluation, it would be convincing to add Equiformer V2 as a baseline to the Ta-V-Cr-W system. This should be relatively straightforward, and I am interested in understanding the relative performance of this model.
Summary
In this work, the authors proposed higher-rank irreducible Cartesian Tensor Product, and explored its usage in equivariant neural networks design in scientific applications such as molecular modeling. The authors firstly prove that irreducible Cartesian Tensor Product is equivariant to O(3) group, and further show that higher-rank (e.g., > 2) operations can be used in an efficient way for models using many-body interactions. Experiments are conducted to demonstrate the effectiveness of the proposed approach.
Strengths
1. The problem this work aims to tackle is of great significance in real-world scientific applications. 2. The proposed approach is interesting and can potentially improve a new class of equivariant neural networks for crucial tasks. 3. The paper is easy to follow.
Weaknesses
1. **The motivation of this work needs to be better explained and presented**. As stated in the Introduction and Related Works, the major disadvantage of spherical tensors is computationally demanding, which motivates the development of Cartesian-Tensor-Product-based approaches. However, the authors do not well discuss the disadvantages of existing Cartesian-Tensor-Product-based approaches (e.g., inefficiency in scaling tensor ranks up), which is necessary as a solid support for the motivation of this work. Besides, the lack of comprehensive discussion and comparisons between this work and existing Cartesian-Tensor-Product-based approaches makes the actual value of this work doubtful for readers who are not familiar with the context. 2. **Experimental results are weak in demonstrating the superiority of the proposed approach**: - Lack of detailed efficiency comparisons: in this work, the authors demonstrate that the proposed irreducible Cartesian Tensor Product can be used for both two-body and many-body feature interaction or equivariant convolutions with better theoretically-proved efficiency. However, there are no comprehensive comparisons covering different operations and also different rank L. - Lack of large-scale experiments: all datasets used in this work (rMD17, 3BPA and Acetylacetone) have limited scales of molecular systems and number of samples. Since the proposed approach is claimed to bring benefits in efficiency, it would be necessary to verify it on larger-scale datasets such as OC20/22. - Lack of experiments on applying iCTP for two-body operations only: MACE is mainly used to compare iCTP and spherical tensor products. However, two-body operations like equivariant feature interaction/equivariant convolution are also widely used in equivariant networks. It would be better to further verify the effectiveness of iCTP on these operations only to demonstrate its generality. Overall, it is of great significance to design more powerful and efficient equivariant networks for real-world applications. However, several issues exist in the current submission. My recommendation is Borderline Accept, and I will carefully read the rebuttal and other reviews to decide whether to decrease or increase my scores.
Questions
1. Could you detailedly explain the difference in computational complexity between iCTP and spherical Tensor Products on both two-body and many-body interactions? 2. Could you comprehensively compare this work with TensorNet and compare the strengths and weaknesses of them?
Rating
6
Confidence
4
Soundness
2
Presentation
3
Contribution
3
Limitations
The authors have discussed the limitations.
Tab. 1: Inference times and memory consumption as a function of L and ν for the 3BPA data set.
All values are obtained by averaging over five independent runs. Best performances are highlighted in bold. Inference time and memory consumption are measured for a batch size of 10. Inference time is reported per structure in ms; memory consumption is provided for the entire batch in GB. | | *L* = 1 | | *L* = 2 | | *L* = 3 | | |:---|:---|:---|:---|:---|:---|:---| | | ICTP | MACE | ICTP | MACE | ICTP | MACE | | Inference times | | | | | | | | *ν* = 1 | **0.76 ± 0.17** | 1.02 ± 0.03 | **0.87 ± 0.18** | 1.38 ± 0.04 | **0.98 ± 0.26** | 1.88 ± 0.03 | | *ν* = 2 | **0.59 ± 0.20** | 1.12 ± 0.03 | **1.03 ± 0.21** | 1.52 ± 0.05 | **1.34 ± 0.08** | 2.0 ± 0.10 | | *ν* = 3 | **0.79 ± 0.22** | 1.23 ± 0.03 | **1.15 ± 0.08** | 1.67 ± 0.03 | **1.85 ± 0.13** | 2.23 ± 0.03 | | *ν* = 4 | **0.94 ± 0.17** | 1.41 ± 0.11 | **1.31 ± 0.21** | 1.83 ± 0.01 | **2.07 ± 0.20** | 2.53 ± 0.01 | | *ν* = 5 | **1.02 ± 0.17** | 1.52 ± 0.08 | **1.72 ± 0.07** | 2.26 ± 0.03 | **3.61 ± 0.02** | OOM | | *ν* = 6 | **1.00 ± 0.07** | 1.77 ± 0.05 | **1.83 ± 0.16** | 27.85 ± 0.01 | **16.76 ± 0.35** | OOM | | Memory consumption | | | | | | | | *ν* = 1 | 0.05 ± 0.00 | **0.04 ± 0.00** | 0.08 ± 0.00 | **0.06 ± 0.00** | 0.21 ± 0.00 | **0.13 ± 0.00** | | *ν* = 2 | 0.05 ± 0.00 | **0.04 ± 0.00** | 0.08 ± 0.00 | **0.07 ± 0.00** | 0.28 ± 0.09 | **0.13 ± 0.00** | | *ν* = 3 | 0.05 ± 0.00 | **0.04 ± 0.00** | 0.10 ± 0.00 | **0.08 ± 0.00** | 0.51 ± 0.03 | **0.23 ± 0.00** | | *ν* = 4 | **0.05 ± 0.00** | **0.05 ± 0.00** | **0.18 ± 0.08** | 0.30 ± 0.00 | **1.07 ± 0.10** | 4.16 ± 0.00 | | *ν* = 5 | **0.05 ± 0.00** | 0.07 ± 0.00 | **0.35 ± 0.07** | 3.18 ± 0.00 | **5.07 ± 0.02** | OOM | | *ν* = 6 | **0.11 ± 0.09** | 0.22 ± 0.00 | **0.93 ± 0.00** | 50.49 ± 0.00 | **28.48 ± 0.03** | OOM |
Tab. 2: Energy (E, meV) and force (F, meV/Å) RMSEs for the 3BPA data set and ν = 1.
All values are obtained by averaging over five independent runs. Best performances are highlighted in bold. Inference time and memory consumption are measured for a batch size of 100. Inference time is reported per structure in ms; memory consumption is provided for the entire batch in GB. | | | ICTP (*L* = 2) | MACE (*L* = 2) | |:-------------------------------|:---:|-------------------:|-----------------:| | 300 K | E | **12.90 ± 1.06** | **13.50 ± 1.71** | | | F | **29.90 ± 0.25** | **30.18 ± 0.38** | | 600 K | E | **29.97 ± 0.94** | **31.32 ± 2.16** | | | F | **62.80 ± 0.45** | **63.04 ± 0.73** | | 1200 K | E | **81.03 ± 1.64** | **81.54 ± 2.02** | | | F | **146.96 ± 1.30** | 149.44 ± 1.94 | | Dihedral slices | E | **22.84 ± 2.96** | 28.08 ± 4.04 | | | F | **48.82 ± 5.25** | **49.62 ± 2.92** | | Inference time | | **2.62 ± 0.02** | 2.96 ± 0.06 | | Memory consumption | | 32.57 ± 0.00 | **23.32 ± 0.00** |
Tab. 3: Energy (E, meV) and force (F, eV/Å) RMSEs for Ta-V-Cr-W subsystems.
Results are obtained by averaging over 10 independent runs. Best performances are highlighted in bold. Inference time and memory consumption are measured for a batch size of 50. Inference time is reported per atom in μs; memory consumption is provided for the entire batch in GB. | Subsystem | | ICTP (*L* = 2) | ICTP (*L* = 1) | ICTP (*L* = 0) | MACE (*L* = 2) | MACE (*L* = 1) | MACE (*L* = 0) | ICTP (*L* = 2, *ν* = 2) | MTP | GM-NN | EAM | |:---|:--:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:| | TaV | E | **1.02 ± 0.27** | **1.21 ± 0.54** | 1.65 ± 1.06 | 1.72 ± 0.67 | 1.76 ± 0.53 | 2.24 ± 1.34 | **1.24 ± 0.50** | 1.94 | 1.54 | 32.0 | | | F | **0.020 ± 0.002** | 0.022 ± 0.002 | 0.024 ± 0.002 | **0.019 ± 0.002** | **0.020 ± 0.003** | 0.022 ± 0.002 | 0.023 ± 0.002 | 0.050 | 0.029 | 0.404 | | TaCr | E | **1.81 ± 0.29** | **1.94 ± 0.23** | 2.13 ± 0.19 | 3.26 ± 0.42 | 3.31 ± 0.44 | 4.18 ± 0.56 | 2.4 ± 0.33 | 3.26 | 2.98 | 43.6 | | | F | **0.025 ± 0.007** | **0.024 ± 0.006** | 0.027 ± 0.005 | 0.029 ± 0.01 | **0.026 ± 0.007** | 0.028 ± 0.007 | **0.026 ± 0.006** | 0.057 | 0.038 | 0.343 | | TaW | E | **1.75 ± 0.11** | 1.87 ± 0.14 | 2.45 ± 0.31 | 2.73 ± 0.53 | 3.21 ± 0.55 | 3.57 ± 0.48 | 2.19 ± 0.54 | 2.72 | 2.99 | 44.8 | | | F | **0.017 ± 0.002** | **0.018 ± 0.002** | 0.020 ± 0.002 | **0.017 ± 0.002** | **0.018 ± 0.002** | 0.019 ± 0.002 | **0.018 ± 0.002** | 0.038 | 0.025 | 0.248 | | VCr | E | **1.74 ± 1.2** | 2.52 ± 2.43 | **2.13 ± 1.24** | **2.19 ± 0.78** | 2.82 ± 1.28 | 3.11 ± 1.42 | **1.89 ± 1.27** | **2.29** | 2.82 | 44.8 | | | F | **0.016 ± 0.002** | 0.018 ± 0.001 | 0.019 ± 0.001 | **0.016 ± 0.001** | **0.017 ± 0.001** | 0.018 ± 0.002 | 0.019 ± 0.001 | 0.036 | 0.025 | 0.270 | | VW | E | **1.32 ± 0.2** | **1.46 ± 0.16** | 1.69 ± 0.21 | 1.9 ± 0.19 | 1.94 ± 0.23 | 2.42 ± 0.24 | 1.61 ± 0.16 | 2.50 | 2.00 | 21.3 | | | F | **0.014 ± 0.002** | **0.015 ± 0.002** | 0.018 ± 0.003 | **0.014 ± 0.002** | **0.015 ± 0.002** | 0.017 ± 0.002 | 0.016 ± 0.002 | 0.037 | 0.023 | 0.292 | | CrW | E | **2.18 ± 0.93** | **2.45 ± 1.53** | 2.76 ± 1.15 | **2.31 ± 1.18** | 2.84 ± 0.98 | 4.14 ± 1.38 | 3.12 ± 1.90 | 4.35 | 2.87 | 23.4 | | | F | **0.018 ± 0.004** | **0.020 ± 0.005** | 0.024 ± 0.008 | **0.020 ± 0.009** | **0.019 ± 0.006** | 0.023 ± 0.007 | 0.022 ± 0.006 | 0.041 | 0.029 | 0.248 | | TaVCr | E | **0.79 ± 0.08** | 0.92 ± 0.17 | 1.00 ± 0.24 | 2.26 ± 0.54 | 2.71 ± 0.66 | 3.92 ± 0.77 | 0.97 ± 0.13 | 2.43 | 1.97 | 34.1 | | | F | 0.027 ± 0.001 | 0.029 ± 0.002 | 0.033 ± 0.002 | **0.023 ± 0.002** | **0.024 ± 0.001** | 0.028 ± 0.001 | 0.031 ± 0.002 | 0.054 | 0.045 | 0.313 | | TaVW | E | **1.00 ± 0.2** | **0.98 ± 0.18** | 1.26 ± 0.23 | 1.8 ± 0.35 | 1.97 ± 0.44 | 2.29 ± 0.86 | **0.95 ± 0.25** | 1.67 | 1.70 | 39.6 | | | F | **0.021 ± 0.001** | 0.022 ± 0.001 | 0.025 ± 0.001 | **0.021 ± 0.002** | 0.023 ± 0.001 | 0.026 ± 0.001 | 0.023 ± 0.001 | 0.043 | 0.034 | 0.321 | | TaCrW | E | **1.16 ± 0.15** | **1.28 ± 0.13** | 1.58 ± 0.29 | 1.67 ± 0.38 | 1.48 ± 0.50 | 2.08 ± 0.57 | **1.24 ± 0.11** | 2.08 | 2.19 | 23.6 | | | F | **0.022 ± 0.001** | 0.024 ± 0.001 | 0.027 ± 0.001 | 0.028 ± 0.002 | 0.030 ± 0.002 | 0.033 ± 0.002 | 0.026 ± 0.001 | 0.051 | 0.039 | 0.327 | | VCrW | E | **1.00 ± 0.16** | **1.07 ± 0.14** | 1.37 ± 0.13 | 1.97 ± 0.5 | 2.21 ± 0.42 | 2.86 ± 0.64 | **1.10 ± 0.14** | 1.37 | 1.94 | 19.4 | | | F | **0.018 ± 0.001** | 0.019 ± 0.001 | 0.022 ± 0.001 | **0.017 ± 0.001** | 0.019 ± 0.001 | 0.021 ± 0.001 | 0.020 ± 0.001 | 0.040 | 0.031 | 0.314 | | TaVCrW (0 K) | E | **1.22 ± 0.07** | 1.30 ± 0.1 | 1.48 ± 0.16 | 2.26 ± 0.55 | 2.48 ± 0.46 | 3.60 ± 0.54 | **1.33 ± 0.17** | 2.09 | 2.16 | 50.8 | | | F | **0.021 ± 0.002** | **0.022 ± 0.002** | 0.025 ± 0.002 | **0.022 ± 0.001** | 0.023 ± 0.002 | 0.027 ± 0.001 | 0.024 ± 0.002 | 0.049 | 0.037 | 0.488 | | TaVCrW (2500 K) | E | **1.63 ± 0.07** | 1.74 ± 0.11 | 2.09 ± 0.09 | 2.22 ± 0.48 | 2.34 ± 0.59 | 3.68 ± 0.70 | 2.06 ± 0.09 | 2.40 | 2.67 | 59.4 | | | F | **0.116 ± 0.002** | 0.121 ± 0.002 | 0.141 ± 0.003 | **0.119 ± 0.007** | 0.126 ± 0.006 | 0.150 ± 0.003 | 0.140 ± 0.002 | 0.156 | 0.179 | 0.521 | | Overall | E | **1.38 ± 0.09** | 1.56 ± 0.21 | 1.80 ± 0.18 | 2.19 ± 0.31 | 2.42 ± 0.31 | 3.17 ± 0.28 | 1.67 ± 0.21 | 2.43 | 2.32 | 37.14 | | | F | **0.028 ± 0.001** | **0.029 ± 0.001** | 0.034 ± 0.001 | **0.029 ± 0.001** | 0.030 ± 0.001 | 0.034 ± 0.001 | 0.032 ± 0.001 | 0.054 | 0.043 | 0.443 | | Inference time | | 51.78 ± 1.18 | 25.09 ± 0.02 | 14.59 ± 0.01 | 29.48 ± 0.23 | 15.37 ± 0.04 | 4.43 ± 0.00 | 14.97 ± 0.09 | 17.57 | 7.25 | 0.50 | | Memory consumption | | 36.78 ± 0.00 | 16.93 ± 0.00 | 8.48 ± 0.00 | 28.82 ± 0.00 | 13.87 ± 0.00 | 5.91 ± 0.00 | 13.15 ± 0.00 | – | – | – |
Rebuttal reply
I would like to thank the authors for their rebuttal. They have addressed my concerns satisfactorily, providing extensive clarifications, particularly on how TensorNet and CACE are related to the present work, a computational complexity comparison to MACE, and additional experiments. Furthermore, they provide good additional results, both in terms of inference times and in terms of accuracy on a more diverse dataset. Given this, and after reading other reviewers' impressions and how they are addressed by the authors, I rise my score.
Decision
Accept (poster)