Surrogate Modeling for the Design of Optimal Lattice Structures using Tensor Completion

TL;DR

Design of lattice-structure materials suffers from an exponential design space and nonuniform training data. The paper formulates this as a tensor completion problem and applies CPD-based and neural tensor completion methods (e.g., CPD-S and NeAT) trained with MAE via Adam, including ensemble strategies. It finds that tensor completion can outperform biased-sampling baselines by around $5\%$ in $R^2$ while remaining competitive under uniform sampling, indicating robustness to nonuniform data. This approach provides a practical surrogate to accelerate lattice-design exploration, enabling efficient screening of high-performance structures in realistic experimental settings.

Abstract

When designing new materials, it is often necessary to design a material with specific desired properties. Unfortunately, as new design variables are added, the search space grows exponentially, which makes synthesizing and validating the properties of each material very impractical and time-consuming. In this work, we focus on the design of optimal lattice structures with regard to mechanical performance. Computational approaches, including the use of machine learning (ML) methods, have shown improved success in accelerating materials design. However, these ML methods are still lacking in scenarios when training data (i.e. experimentally validated materials) come from a non-uniformly random sampling across the design space. For example, an experimentalist might synthesize and validate certain materials more frequently because of convenience. For this reason, we suggest the use of tensor completion as a surrogate model to accelerate the design of materials in these atypical supervised learning scenarios. In our experiments, we show that tensor completion is superior to classic ML methods such as Gaussian Process and XGBoost with biased sampling of the search space, with around 5\% increased $R^2$. Furthermore, tensor completion still gives comparable performance with a uniformly random sampling of the entire search space.

Figures (4)

• Figure 1: We present various material design problems as instances of tensor completion, in order to abstract away the various design components and use tensor methods to infer the entire search space.
• Figure 2: Here we visualize the training process, where we iteratively update our factor matrices to produce a better tensor reconstruction. The gray regions in the factor matrices and predicted tensor correspond to predicted values for which we do not have the ground truth during training.
• Figure 3: In (a) we display a parity plot for predicting $E$ and $\tilde{E}$ simultaneously. We use 100 train and 170 test values. In (b) we display the $R^2$ for a uniform random sampling of different training sizes. We show the mean and standard deviation on the test set, using 5 iterations for each train size.
• Figure 4: In (a) we visualize how the search space might be sampled in a biased manner. In (b), we show results for a biased-random sampling of the search space. We display the average and standard deviation of the $R^2$ for 5 iterations. Going from experiment 1 to 10, we decrease the bias in sampling by making the range smaller. The exact setup is provided in the appendix section in Table \ref{['biased_sampling_trainset']}.