Tensor Methods: A Unified and Interpretable Approach for Material Design

TL;DR

The paper proposes tensor completion, particularly Canonical Polyadic Decomposition (CPD), as an interpretable surrogate modeling approach for material design to navigate large design spaces and handle non-uniform sampling. It demonstrates that CPD-based models yield interpretable tensor factors that align with underlying physics and can rediscover known phenomena, while maintaining competitive predictive performance against traditional ML baselines. Across three design datasets, tensor methods show robust generalization in biased sampling scenarios and, in some cases, outperform standard models in out-of-distribution regions. The work highlights the trade-offs between interpretability and predictive power and provides empirical evidence that tensor factors can serve as a practical tool for experimentalists to identify patterns and potentially novel materials. Overall, tensor completion emerges as a practical, interpretable, and generalizable surrogate modeling paradigm for material design, with public code and clear implications for biased-sampling environments.

Abstract

When designing new materials, it is often necessary to tailor the material design (with respect to its design parameters) to have some desired properties (e.g. Young's modulus). As the set of design parameters grow, the search space grows exponentially, making the actual synthesis and evaluation of all material combinations virtually impossible. Even using traditional computational methods such as Finite Element Analysis becomes too computationally heavy to search the design space. Recent methods use machine learning (ML) surrogate models to more efficiently determine optimal material designs; unfortunately, these methods often (i) are notoriously difficult to interpret and (ii) under perform when the training data comes from a non-uniform sampling of the design space. We suggest the use of tensor completion methods as an all-in-one approach for interpretability and predictions. We observe classical tensor methods are able to compete with traditional ML in predictions, with the added benefit of their interpretable tensor factors (which are given completely for free, as a result of the prediction). In our experiments, we are able to rediscover physical phenomena via the tensor factors, indicating that our predictions are aligned with the true underlying physics of the problem. This also means these tensor factors could be used by experimentalists to identify potentially novel patterns, given we are able to rediscover existing ones. We also study the effects of both types of surrogate models when we encounter training data from a non-uniform sampling of the design space. We observe more specialized tensor methods that can give better generalization in these non-uniforms sampling scenarios. We find the best generalization comes from a tensor model, which is able to improve upon the baseline ML methods by up to 5% on aggregate $R^2$, and halve the error in some out of distribution regions.

Figures (11)

• 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. We first visualize how we are able to convert various material design optimization problems (designing an optimal lattice structure, 3D printed structures, and electrospinning configurations for nanofiber development) into instances of tensor completion problems. Then we depict a biased sampling of the design space (which is what a surrogate model's training data might consist of in practice), resulting from oversampling from low-cost designs and undersampling from high-cost designs (in this visualization "cost" is measured as FEA simulation runtimes). Furthermore, we display a Canonical Polyadic Decomposition (CPD) tensor model, which generates interpretable tensor factors automatically as a result of its predictions of the design space. The parts of the images depicting the actual lattice structure designs gongora2024accelerating and the crossed barrel designs gongora2020bayesian come from their original papers. Also please note the histogram representing FEA simulation times is synthetic, and inspired by the actual distribution of FEA runtimes from the original paper gongora2024accelerating (the real histogram can be found in the original paper's supplementary figures).
• Figure 2: The design spaces and outcomes of interest for the datasets used in our experiments. We also display the total possible design combinations (Total) and the number of observed combinations (Observed) in each dataset, from the set of design parameter values. For the lattice structures dataset, design are solely parameterized by the unit cell (UC): Geometry, Thickness, and the length in the X, Y, and Z directions. For the crossed barrel dataset, the designs are solely parameterized by the struts: number of struts, $\theta$ for the struts' twist, strut radius, and strut thickness. For the Cogni-e-Spin dataset, designs are parameterized by the electrospinning configurations: solution concentration, voltage, flow rate, tip-to-collector distance, and the polymer.
• Figure 3: For the lattice structures dataset, we visualize CPD (rank 3) factors to study any trends in the design parameters. In this experiment, we use only 25% of training values and display the factors generated using only this training data in (a) & (c). We also show the true distribution with respect to these design parameter values in (b) & (d), along with a vertical line to represent the mean value.. For the Unit Cell Z length, we notice a distinct increase in component magnitude with an increasing length (a). We can associate this with a correlation between the Unit Cell Z length and $\tilde{E}$ (b), which we also observe from the true distribution of the data. This finding is also observed in the dataset's original paper gongora2024accelerating. We compare this with the Unit Cell X length parameter (c), which does not express a clear observable trend, as we can see from the full dataset's distribution with respect to the parameter values (d).
• Figure 4: For the crossed barrel dataset, we display the CPD (rank 4) factors to study any trends in the design parameters. Since this dataset is more complicated than the lattice dataset, it is difficult to find any one design parameters' effect on the outcome (we can do a rank 1 decomposition, but then prediction performance suffers). Instead we try to identify any clusters of materials that may be interesting. We plot the CPD normalized components for the $\theta$ and r design parameters in (a) and (b), and we can easily identify a cluster corresponding to component 1 (the blue bars). We highlight in yellow the design parameter values that have a high expression of component 1, and the plot the outcome of interest (toughness) for this cluster, compared with the remaining values outside this cluster, in plot (c). We observe that this cluster is associated with the designs with some of the highest toughness values.
• Figure 5: For the Cogni-e-Spin dataset, we display the a CPD (rank 3) factors. It is unclear whether any explicit trends exist, like with the lattice and crossed barrel datasets. However we can still get a "feature importance" from the tensor factors. We observe a distinct fluctuation in factor magnitudes for the solution concentration, whereas the tip-to-collector distance has similar factor magnitudes. This is reflected in the real distribution, as we see more variance in the outcome (fiber diameter) as we vary the solution concentration, and less variability when we vary the tip-to-collector distance.