🔬 materials science

Discovering new photovoltaics using optimal transport theory

This paper introduces the Fused Gromov-Wasserstein (FGW) metric, an optimal transport-based approach for quantifying material similarity by balancing structural and compositional features, which successfully identified seven previously unexplored high-efficiency photovoltaic candidates, including the stable Cs $_5$ Sb $_8$ , with minimal training data.

Original authors: Matthew A. H. Walker, Zibo Zhou, Junayd Ul Islam, Keith T. Butler

Published 2026-02-27

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CC BY 4.0

Original authors: Matthew A. H. Walker, Zibo Zhou, Junayd Ul Islam, Keith T. Butler

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master chef trying to invent a new, delicious dish. You know that Pizza is amazing, and you know that Tacos are great. You want to find a new recipe that tastes like a mix of both, but you don't want to just copy them exactly. You need a way to say, "This new dish is similar to a pizza in some ways and similar to a taco in others."

This is exactly the problem scientists face when looking for new solar panels (photovoltaics). They know which chemical materials make great solar panels (like Silicon or Perovskites), but they need to find new materials that are similar enough to work well, but different enough to be new discoveries.

Here is how this paper solves that problem, explained simply:

1. The Problem: How do you measure "Similarity"?

For a long time, scientists tried to find new materials by looking at their ingredients (chemical composition) or their shape (crystal structure).

The Ingredient Approach: "This material has Lead and Iodine, just like the good solar panel, so it must be good."
The Shape Approach: "This material is built in a cube, just like the good one, so it must be good."

The problem is that sometimes a material has the right ingredients but the wrong shape, or the right shape but the wrong ingredients. Scientists struggled to create a single "score" that balanced both. It's like trying to rate a car based only on its engine or only on its tires, but not both.

2. The Solution: The "Optimal Transport" Map

The authors used a mathematical tool called Optimal Transport Theory. Think of this as a logistics company trying to move dirt from a pile (Material A) to fill a hole (Material B) in the most efficient way possible.

They applied this to atoms:

Imagine the atoms in a solar material are like people at a party.
Some people are standing close together (structural bonds).
Some people are wearing specific colored shirts (chemical elements).
The FGW (Fused Gromov-Wasserstein) method is like a super-smart bouncer who looks at two different parties. It asks: "How many people do I need to move from Party A to Party B to make them look the same? And how much 'effort' does it take to move them, considering both their shirt colors and who they are standing next to?"

This creates a distance score. A low score means the two materials are very similar (easy to transform one into the other). A high score means they are very different.

3. The "Magic" of Minimal Training

Usually, to teach a computer to recognize good solar materials, you need to feed it millions of examples (like training a dog with thousands of treats). This is expensive and slow.

However, this new method is like a genius with a strong intuition. Because the math is built on the fundamental rules of how atoms connect (the "inductive bias"), it doesn't need millions of examples. It only needed about 700 examples to learn the pattern.

The authors tested this against a super-computer-trained AI (a Graph Neural Network) that had studied over 1 million materials. Surprisingly, their simple, low-data method performed just as well as the massive AI!

4. The Treasure Hunt

Once they had their "similarity ruler," they went on a treasure hunt:

The Seeds: They picked the best-known solar materials (the "Seeds").
The Search: They looked through a giant database of 155,000 known materials (The Materials Project).
The Filter: They used their ruler to find materials that were "close" to the seeds but had never been tested as solar panels before.
The Validation: They used powerful supercomputers to double-check the physics of the top candidates.

5. The Discovery

The hunt was a success. They found 7 new materials that look like they could be incredibly efficient solar panels.

The star of the show is a material called Cs5Sb8 (Cesium Antimony).

Prediction: It is predicted to be over 30% efficient (which is very high for solar).
Stability: It is chemically stable (it won't fall apart).
Novelty: No one had ever thought to use this specific combination of atoms for solar power before. It's a completely new recipe found by looking at the "flavor profile" of old recipes.

The Big Picture

This paper shows that you don't always need a massive, expensive AI to discover new science. By using a clever mathematical framework that respects both the ingredients and the structure of materials, scientists can find hidden gems in the data that traditional methods miss.

It's like finding a new, perfect recipe for a solar panel not by guessing, but by mathematically understanding how the "flavors" of the universe mix together.

1. Problem Statement

The discovery of new photovoltaic (PV) materials traditionally relies on chemical and structural analogy (e.g., moving from Se to Si to GaAs). However, defining "similarity" between materials for algorithmic search remains an open challenge.

The Gap: Existing methods struggle to balance compositional similarity (chemical elements) and structural similarity (crystal lattice geometry) within a single, unified metric.
Limitations of Current Approaches:
- Composition-only: Vectors based on elemental properties (e.g., Magpie embeddings) ignore crystal structure.
- Structure-only: Geometric matching (e.g., StructureMatcher) fails to quantify similarity for non-identical structures.
- Deep Learning: Graph Neural Networks (GNNs) require massive datasets (often $>10^6$ materials) to train effectively, which is prohibitive for specific tasks like PV screening where labeled data is scarce.

2. Methodology

The authors propose adapting the Fused Gromov-Wasserstein (FGW) distance, a concept from Optimal Transport theory, to measure the dissimilarity between crystalline materials.

Core Concept: Fused Gromov-Wasserstein (FGW)

FGW maps atoms (nodes) and atomic pairs (edges) between two material graphs to find the "optimal transport" plan that minimizes the cost of transformation. It balances two costs:

Feature Cost ( $M_{AB}$ ): The distance between feature vectors of atoms (e.g., chemical composition).
Structural Cost ( $L(C_A, C_B)$ ): The difference in the relational structure (connectivity/distances) between atoms in the two materials.

The distance is controlled by a hyperparameter $\alpha \in [0, 1]$ :

$\alpha = 0$ : Purely compositional (Wasserstein distance).
$\alpha = 1$ : Purely structural (Gromov-Wasserstein distance).
$0 < \alpha < 1$ : A fused metric balancing both.

Implementation Details

Graph Representation: Materials are represented as graphs using StructureGraph from Pymatgen, where nodes are atoms and edges are determined by the CrystalNN (Voronoi) strategy.
Feature Vectors: The study tested various atom embeddings, including:
- One-hot vectors.
- Random distributed vectors.
- Empirical vectors (Magpie, Oliynik).
- Learned embeddings (SkipAtom, CrystaLLM).
Target Metric: The FGW distance was correlated with Spectroscopic Limited Maximum Efficiency (SLME), a heuristic for PV power conversion efficiency that accounts for bandgap, absorption, and thickness.
Workflow:
1. Hyperparameter Tuning: Optimized $\alpha$ and feature choices on a small dataset (695 materials) by minimizing Binary Cross-Entropy (BCE) loss between FGW distances and SLME differences.
2. Clustering: Used $k$ -medoid clustering to group materials into chemical-structural families.
3. Discovery Campaign: Selected "seed" materials (highest SLME in each cluster) and searched the Materials Project database (155k materials) for candidates with low FGW distances.
4. Validation: Filtered candidates by bandgap (0–2.5 eV), stability, and dimensionality, then validated top candidates using Hybrid DFT (HSE06) calculations.

3. Key Contributions

Novel Metric for Materials Science: Successfully adapted FGW, previously used in computer vision and NLP, to quantify the similarity of crystalline materials by fusing composition and structure.
Data Efficiency: Demonstrated that FGW achieves high predictive accuracy with minimal training data (only ~700 materials) compared to state-of-the-art Graph Neural Networks (GNNs) that require millions of training examples.
Inductive Bias: Showed that strong inductive biases (via chemical embeddings and structural matrices) allow for effective materials exploration without the computational cost of training large deep learning models.
Open Source Tools: Released ChemFGW, a Python package adapting the FGW algorithm for chemical graphs.

4. Results

Performance Benchmarking

Hyperparameter Optimization: The lowest BCE loss occurred at $\alpha \in (0, 1)$ , confirming that a combination of compositional and structural costs is essential for predicting PV activity. Purely compositional ( $\alpha=0$ ) or structural ( $\alpha=1$ ) approaches performed poorly.
Comparison with Baselines:
- FGW methods (even with simple one-hot vectors) achieved Root Mean Square Error (RMSE) comparable to MACE, a state-of-the-art equivariant GNN pre-trained on $10^6$ materials.
- FGW significantly outperformed composition-only baselines (Magpie) and standard structural kernels (SOAP).

Materials Discovery

Clustering: The method successfully clustered 695 materials into 12 distinct groups, identifying diverse "seed" materials with high SLME (>30%).
Local Enrichment: Analysis showed that materials near high-performing seeds in FGW space had significantly higher SLMEs than the dataset average, validating the metric's ability to navigate the "PV landscape."
Candidate Identification:
- Screened 155,000 materials from the Materials Project.
- Identified 7 previously unexplored high-efficiency PV candidates after DFT validation.
- Top Discovery: Cs $_5$ Sb $_8$ .
  - Predicted SLME: >30%.
  - Stability: Thermodynamically stable.
  - Novelty: No prior literature exists on this material as a PV absorber.
- Other promising candidates included CaAgAs (SLME ~29.5%) and Tl $_4$ GeSe $_4$ (SLME ~22.2%).

5. Significance

This work establishes FGW as a powerful, low-cost alternative to data-hungry deep learning for materials discovery.

Scalability: It enables high-fidelity similarity searches without the need for massive pre-training datasets.
Generalizability: The approach successfully identified complex, non-trivial chemical substitutions (e.g., Bi $_{20}$ (PtI $_{12}$ ) $_3$ derived from BiSeI) that simple similarity rules would miss.
Practical Impact: The discovery of Cs $_5$ Sb $_8$ with a predicted SLME >30% provides a concrete, experimentally viable target for next-generation solar cells, demonstrating the practical utility of optimal transport theory in accelerating the transition to renewable energy.

In summary, the paper proves that by mathematically fusing chemical composition and crystal structure via optimal transport, researchers can efficiently navigate the vast materials space to discover high-performance photovoltaics with minimal computational overhead.