Summary
This paper proposes an application of equivariant GNNs for predicting the stiffness tensor of architected lattice metamaterials. To ensure the validity of physical significance, a layer is proposed to preserve the positive semi-definiteness of the predicted stiffness.
Strengths
1. There is less work focused on studying higher-order tensors beyond first-order tensors, such as coordinates, velocity, and force. Research on a 4th-order tensor represents a new situation for application.
2. This paper try to design a new module to ensure the validity of physical significance, which is inspiring.
3. The figures are pretty.
Weaknesses
1. This work chooses to predict a 4-th ordered tensor – “stiffness tensor”, whose symmetry has been maintain well with existing models like TFN, PAINN. This paper does not propose theoretical innovations in maintaining equivariance, so this methodology and the emphasis on “fourth-order” in abstract are not directly related (the Positive Semi-Definite Layer is based on physical meaning, regardless of whether it is a fourth-order tensor).
2. Since the whole backbone is built with existing MACE layer, the main novelty is to propose the “Positive Semi-Definite Layer”(let us call it “PSD-layer”). About this, here are some questions:
- a) To maintain the equivariance of the total model, we must ensure each layer in such a model is equivariant. But for the PSD-layer, the PSD matrix $A$ is based on $M$, which is created by arrange $n(n+1)/2$ entries of the output of an equivariant model.
- **i**. Is it an equivariant operator to arrange $n(n+1)/2$ entries to get $M$? This requires a formal mathematical proof.
- **ii**. Is it an equivalent operator of the function $\rho$ to turn $M$ into $A$? In the end of Section A.2.2, “If the overall model had been equivariant with respect to vectors in $U$, it will remain equivariant after eigenvalues are made positive” is not trivail, it may also require a formal mathematical proof.
- **iii**. For now, we assume that the previous question (function $\rho$ is equivariant) has a good proof. It is necessary to discuss whether this design will reduce the representational capacity of the entire neural network. Assume that this neural network is a bijection (we consider an equivalence class divided based on different perspectives as an input. But the function $\rho$ may not a bijection, e.g. $(\pm \Lambda) ^2$ will get a same result. The Wigner-D based network build a faith representation of the group, but the induction of function $rho$ will lead to an unfaith representation, it requires more analysis. For example, an analysis for the increased stiffness-based errors of CGC+ve, NNConv+ve, and MACE+ve in Table 1 may be a manifestation of the decline in the ability of network representation.
- b) Let assume the PSD-layer is equivariant, there are some questions about the experiments:
- **i**. In Section 3 (Related Work) and Section 5(Conclusion), Finite element (FE) modelling is mentioned, but the experiment could not find the data comparison (e.g. time cost or accuracy) between your model and FE.
- **ii**. More powerful baselines should be discussed, especially the geometric GNNs like TFN[1], NequIP [2], SCN[3], GMN[4], SE(3)-Transformer [5].
3. Here are some possible typos and recommended symbol modifications:
- a) Almost all the inline formulas in the article lack punctuation at the end.
- b) In Section 2.1, the rotation of the stiffness tensor $R_{ia}R_{jb}C_{abcd}R_{kc}R_{ld}$ may be better to generalize to $n$-th order tensors with the format of changing bases of a tensor($ R_{ia}R_{jb} R_{kc}R_{ld} C_{abcd}$).
- c) In Eq. (2), the Wigner-D matrix generally written in the form $D_{m’,m}^{(l)}(R)$.
- d) In Section 4, the third line in the paragraph “Positive Semi-Definite Layer”, “in line wih”, may be “in line with”.
Reference:
- [1] Tensor field networks: Rotation-and translation-equivariant neural networks for 3d point clouds.
- [2] E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials
- [3] Spherical Channels for Modeling Atomic Interactions
- [4] Equivariant Graph Mechanics Networks with Constraints
- [5] SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks
Confidence
4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.