Tensor Methods: A Unified and Interpretable Approach for Material Design

AI summary

Authors: Shaan Pakala (UC Riverside), Aldair E. Gongora (Lawrence Livermore National Laboratory), Brian Giera (Lawrence Livermore National Laboratory), Evangelos E. Papalexakis (UC Riverside) Venue: Proceedings of the 32nd ACM SIGKDD Conference on Knowledge Discovery and Data Mining V.2 (KDD '26), Jeju Island, Republic of Korea, August 09–13, 2026 arXiv: 2602.10392v3 [cs.LG] Code: https://github.com/shaanpakala/Tensor-Methods-for-Material-Design

Overview

• Research area: Machine learning surrogate modeling for materials design, specifically tensor decomposition / tensor completion as an alternative to traditional ML regression models.
• Technical level: Intermediate. The core idea (predicting missing entries of a grid of design combinations) is intuitive, but some familiarity with tensor decomposition, low-rank assumptions, and regression metrics helps.
• Scope: The paper proposes tensor completion as a single method that both predicts material properties and produces interpretable factors for free, and benchmarks this against six traditional ML baselines across three material design datasets under uniform and non-uniform training-data sampling.

What This Paper Is About

When engineers want a material with a target property (for example an optimal Young's Modulus), the number of possible design combinations grows exponentially with each added design variable, so exhaustively simulating or synthesizing every design with tools like Finite Element Analysis (FEA) is impractical. Machine learning surrogate models speed up this search, but the authors argue they have two persistent weaknesses: they are hard to interpret, and they tend to overfit when training data comes from a non-uniform sampling of the design space, which is what happens in practice. The paper's goal is to show that tensor completion methods can match traditional ML on prediction accuracy while automatically yielding interpretable factors that reflect the underlying physics.

Key Contributions

1. Interpretability of tensor methods: The authors study whether tensor factors can rediscover known physical phenomena in the data, and propose that experimentalists could use the same mechanism to surface potentially novel patterns. They compare this to traditional ML, where interpretation requires an expensive post-hoc tool such as SHAP.
2. Non-uniform sampling of training data: They investigate how both traditional ML and tensor-based surrogates behave when training data is heavily sampled from one region of the design space and sparsely sampled elsewhere—a setting the paper notes has very limited prior study.
3. Tensor completion for handling non-uniform sampling: They show cases where tensor completion generalizes better than traditional ML methods in that biased-sampling regime.
4. Benchmarking and dataset variety: They benchmark tensor completion's utility as a surrogate model across three material design scenarios—optimal lattice structures, 3D printed structures optimized for toughness, and electrospinning configurations for nanofiber production—and release their code publicly.

Main Findings

• Tensor models are competitive under uniform sampling: With an 80%–20% train-test split, tensor models are "definitely comparable" to traditional ML models. On the Lattice Structures dataset, CPD reached R² 0.99 ± 0.0, matching GP (0.99 ± 0.0) and XGBoost (0.99 ± 0.0). The paper states that traditional ML methods, mainly GP and CatBoost, generally had the best performance, and that no single model was definitively best across all datasets.
• Best generalization under biased sampling comes from a tensor model: The paper reports that the best generalization came from a tensor model, improving upon baseline ML methods by up to 5% on aggregate R² and halving the error in some out-of-distribution sections of the search space.
• CoSTCo led aggregate results on two of three datasets under biased sampling: CoSTCo achieved R² 0.84 ± 0.0 on Lattice Structures (versus GP at 0.79 ± 0.1 and RF at 0.77 ± 0.0) and R² 0.43 ± 0.1 on Cogni-e-Spin (versus RF and XGBoost at 0.38 ± 0.1). On Crossed Barrel, traditional methods led, with GP at 0.68 ± 0.0 and the best tensor model CPD-S at 0.56 ± 0.1.
• Traditional ML overfits inside the heavily sampled region: Looking at disaggregated MAE by subsection of the design space, the paper reports that traditional ML methods outperform tensor methods within the heavily sampled region but overfit there, while tensor methods generalize better outside it.
• Significance testing: t-tests comparing the best tensor model per dataset against each baseline produced significant results for Lattice Structures against XGBoost (t = 2.74, p = 0.01), CatBoost (t = 5.90, p = 0.00), and RF (t = 2.48, p = 0.02), and for Cogni-e-Spin against GP (t = 4.10, p = 0.00). For Crossed Barrel, all four t-values were negative and significant, favoring the baselines.
• Physics rediscovered via tensor factors: A rank-3 CPD on the Lattice Structures dataset (using 25% of training values) showed component magnitude increasing with UC_Z length, matching the true distribution and a finding from the dataset's original paper, which had used a GP with SHAP. The UC_X length parameter showed no clear trend in either the factors or the true distribution.
• High-toughness design cluster found in Crossed Barrel: A rank-4 CPD trained on 80% of the observed data (80% of 600 observed samples = 480) identified components with increasing expression at high θ (strut twist angle) and low r (strut radius); designs expressing that component corresponded to some of the highest toughness values in the true distribution. The authors connect this to the physics of hollow struts and to prior literature, and note that the relationship is not simply monotonic—pushing θ up and r down to extremes can produce structures that are too stiff or that shatter.
• Low-rank structure varies by dataset: For the Lattice dataset, CPD ranks from 3 to 24 all gave excellent performance (>0.95 R²), indicating a strong low-rank structure. Crossed Barrel showed a sweet spot between ranks 4 and 10. Cogni-e-Spin showed a looser adherence to the low-rank assumption, where CPD performed poorly but a smoothness constraint (CPD-S) could alleviate this. CoSTCo was very robust to rank selection.
• Factor reproducibility: Across 10 random initializations, pairwise Factor Match Scores (FMS) were high for Lattice (0.99 ± 0.0 for train split, 0.83 ± 0.1 for factors, 0.89 ± 0.1 for both), moderate for Crossed Barrel (0.91 ± 0.1, 0.66 ± 0.1, 0.63 ± 0.1), and poor for Cogni-e-Spin (0.64 ± 0.1, 0.42 ± 0.1, 0.36 ± 0.1).
• Interpretability survives biased sampling on two datasets: Comparing rank-3 CPD factors from uniform versus non-uniform training data gave FMS of 0.72 ± 0.04 for Lattice, 0.65 ± 0.06 for Crossed Barrel, and 0.38 ± 0.04 for Cogni-e-Spin.
• Conclusion text not fully available: The supplied content ends partway into the Conclusion section, so the paper's closing statements are not reported here beyond what the abstract and body contain.

Methodology in Plain English

The researchers treat material design as a grid. If a design has a handful of variables—say, unit cell geometry, thickness, and lengths along three axes—you can imagine a multidimensional table where every cell is one combination of design variables and the cell's value is the property you care about (such as Young's Modulus or toughness). Experiments or simulations fill in some cells; the rest are empty. Tensor completion is the task of filling in the empty cells.

The specific tool is the Canonical Polyadic Decomposition (CPD), which expresses that multidimensional table as a sum of rank-one pieces, each piece being the outer product of one vector per design variable. Predicting a missing cell is then just summing, over all ranks, the product of the corresponding entries from each of those vectors. During training, the factor vectors are randomly initialized and then updated by gradient descent with an MSE loss and the Adam optimizer, using only the observed cells as supervision. Because the vectors are jointly used to make predictions, they come out of the process already available—nothing extra has to be trained to interpret them.

The authors compare four tensor models: CPD, CPD-S (which imposes smoothness constraints on some factors), NeAT, and CoSTCo. CoSTCo is a neural tensor completion method; because it is significantly more sensitive to initialization, the authors generate several random sets of initial tensor factors and aggregate them with a Convolutional Neural Network for stability. On the baseline side they use Linear Regression, Gaussian Process, Random Forest, XGBoost, CatBoost, and a Multi-Layer Perceptron, mostly implemented with Scikit-Learn (except XGBoost and CatBoost). Performance is measured with R², MAE, RMSE, and MAPE, and factor similarity is measured with the Factor Match Score implemented in TensorLy.

Two evaluation scenarios are used: a standard setting where training data is drawn uniformly at random, and a biased setting where training data is drawn heavily from a small subsection of the design space defined by two design parameters and only lightly from the rest. Most reported results are means and standard deviations over 10 iterations; the disaggregated error visualization uses 5 iterations, as does the rank-versus-performance study.

Data comes from three sources: the Lattice Structures dataset (270 total and 270 observed combinations, generated with FEA), the Crossed Barrel dataset (1188 total, 600 observed, generated by experimentally 3D printing and evaluating designs with an automated robotic system), and the Cogni-e-Spin dataset (1,168,860 total, 288 observed, synthesized from several other studies of electrospinning parameters and resulting fiber diameter).

Why This Matters

• Impact on research: The paper argues that interpretability and prediction accuracy need not be traded off, and that expensive post-hoc explanation tools like SHAP are unnecessary when interpretation is a byproduct of the prediction itself. It also flags non-uniform training data as an under-studied problem in surrogate modeling for materials and provides a template for evaluating it.
• Materials discovery and design: Accelerating the search for lattice structures that hit a target Young's Modulus, or a target ratio of Young's Modulus to mass.
• Impact-absorbing structures: Finding 3D-printed crossed barrel geometries with high mechanical toughness, relevant to energy absorption applications.
• Nanofiber manufacturing: Tuning electrospinning parameters (solution concentration, voltage, flow rate, tip-to-collector distance, polymer) to control fiber diameter.
• Autonomous and self-driving laboratories: The paper situates its work within the broader push toward automated experimentation, including a cited example of an automated robotic system used to produce the Crossed Barrel data.
• Industry relevance: The work is published at a data mining conference with industrial participation, and the code is publicly released, lowering the barrier for practitioners to test tensor completion against their existing ML surrogate pipelines. The authors also note that a domain expert can study these freely produced factors to look for patterns that are not yet in the literature.

Future Directions

• When does the low-rank assumption hold? The three datasets showed very different behavior: strong low-rank structure for Lattice, a rank sweet spot for Crossed Barrel, and weak structure for Cogni-e-Spin. A principled way to predict which regime a new design problem falls into is an open question the results invite.
• Improving tensor methods for weakly low-rank problems: CPD produced an R² of −0.33 on Cogni-e-Spin under uniform sampling and −0.79 under biased sampling, while CPD-S helped. Developing constrained or hybrid tensor models for such datasets is a natural next step.
• Choosing between tensor and traditional ML models adaptively: The best tensor model won on two datasets under biased sampling while GP led on Crossed Barrel, suggesting value in a diagnostic that selects the surrogate family per problem.
• Using factors to guide experiments: The paper proposes that since existing physical phenomena can be rediscovered from the tensor factors, experimentalists could use them to identify novel patterns—a claim that would need prospective experimental validation rather than retrospective confirmation.

Target Audience

Materials scientists and experimentalists who design lattices, printed structures, or electrospun fibers and want an interpretable surrogate model that works with the biased datasets they actually collect; machine learning and data mining researchers interested in tensor completion and low-rank methods as alternatives to standard regression and black-box models; and practitioners building autonomous or self-driving laboratory pipelines who need models that generalize outside the region where most of their data was collected.

Authors’ abstract

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.