Tensor Methods: A Unified and Interpretable Approach for Material Design

2026-02-11

AI Summary

This work addresses the challenge posed by the exponential growth of the material design parameter space with increasing dimensionality, which limits the efficiency, generalizability, and interpretability of conventional simulation and black-box machine learning approaches. To overcome these limitations, the authors propose a surrogate modeling framework based on tensor decomposition and completion that unifies material property prediction and design parameter analysis. The method efficiently identifies optimal designs from non-uniformly sampled data while maintaining high predictive accuracy and intrinsic interpretability. Notably, it automatically uncovers underlying physical laws and novel design patterns through interpretable tensor factors. Experimental results demonstrate a ~50% reduction in out-of-distribution prediction error and an improvement of up to 5% in average R² compared to baseline methods.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• material design
• tensor methods
• surrogate models
• non-uniform sampling
• interpretability

Innovation

• tensor completion
• interpretable machine learning
• material design
• non-uniform sampling
• surrogate modeling