Tensor Methods: A Unified and Interpretable Approach for Material Design

When designing new materials, it is often necessary to tailor the material design to have some desired properties. As the set of material design parameters grows, the search space grows exponentially, making the actual synthesis and evaluation of all combinations of designs virtually impossible. Even using traditional computational methods, such as Finite Element Analysis (FEA), becomes too computationally heavy to search this 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 entire design space. In this work, 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 methods in predictions, with the added benefit of their interpretable tensor factors (which are given for free). In our experiments, we are able to rediscover physical phenomena via the tensor factors, indicating that our predictions are aligned with the physics of the problem. This also means these 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 (traditional ML \& tensor-based) when we encounter training data from a non-uniform sampling of the design space. We observe some more specialized tensor methods that are able to give better generalization in these non-uniform sampling scenarios, due to the low-rank constraint. 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 sections.