Simple Equations Predict Hydrogen Storage in Porous Materials

A major challenge in moving toward a hydrogen-powered future has always been one deceptively simple question: where do we put the hydrogen? Hydrogen is a clean energy carrier with enormous potential, but storing it safely, compactly, and efficiently has proven difficult. A new study from the University of Michigan offers a promising step forward by showing that simple, physics-based equations can accurately predict how much hydrogen certain advanced materials can store—without the need for supercomputers or opaque machine-learning models.

The research focuses on metal-organic frameworks, or MOFs, a class of nanoporous crystalline materials that have attracted global attention for their ability to trap gases. MOFs are already well known in chemistry and materials science, and their importance was underscored when foundational work in this area contributed to a Nobel Prize in Chemistry. What makes this new work stand out is not the discovery of a new MOF, but the realization that just a few physical traits can reliably predict hydrogen storage performance across tens of thousands of these materials.

Why Hydrogen Storage Is So Hard

Hydrogen is extremely light and has a low energy density by volume. Compressing it to high pressures or liquefying it at ultra-low temperatures comes with safety, cost, and engineering challenges. Solid-state storage—where hydrogen molecules adhere to the surface of a material—has long been seen as a safer alternative. This is where MOFs come in.

MOFs are built from metal nodes connected by organic linkers, forming rigid, sponge-like structures. These structures are mostly empty space. In fact, most MOFs are 80% to 90% empty, meaning a single gram can contain an internal surface area roughly equivalent to a football field. Hydrogen molecules cling weakly to these internal surfaces, allowing the material to store large amounts of gas without chemical bonding.

However, not all MOFs perform equally well. Small changes in pore size, shape, and structure can significantly affect how much hydrogen can be stored—and how much of it can actually be released under real-world conditions.

Moving Beyond Supercomputers and Black Boxes

Until now, predicting hydrogen storage capacity has relied heavily on complex molecular simulations, which are accurate but computationally expensive and slow. Screening millions of potential MOFs using these methods is simply impractical. Machine-learning models have helped speed things up, but they often operate as black boxes, offering predictions without clear physical explanations.

The University of Michigan team took a different route. They used physics-informed symbolic regression, a technique that searches through billions of possible mathematical relationships to find the simplest equations that still match the accuracy of advanced simulations. Instead of letting algorithms guess freely, the researchers constrained the models using known physical principles from the start.

The result is a set of transparent, interpretable equations that can be evaluated almost instantly on a standard computer.

The Two Properties That Matter Most

One of the most striking findings of the study is how little information is actually needed to make accurate predictions. Out of seven structural properties examined, just one or two variables accounted for most of the variation in hydrogen storage performance.

The first is void fraction, which measures how much empty space exists inside the MOF compared to its total volume. This single property alone can accurately predict a material’s volumetric hydrogen capacity, or how much hydrogen it can store per liter.

The second is pore volume, which describes the total volume of the internal pores. When pore volume is combined with void fraction, the equations can accurately predict gravimetric capacity, or how much hydrogen can be stored per unit of weight.

Other features, such as surface area or more detailed geometric descriptors, added only minor improvements. This simplicity surprised even the researchers, who expected these traits to be important but did not anticipate that such constrained equations could rival complex machine-learning models in accuracy.

Massive Datasets and Real-World Conditions

The study is built on an extensive dataset of 88,400 MOFs, each characterized by seven key structural properties. For each material, hydrogen storage capacity was calculated under realistic operating conditions: pressures ranging from 100 bar down to 5 bar and a temperature of 77 kelvin, which corresponds to liquid nitrogen cooling.

To ensure the equations were not overfitted, the team validated them against a much larger superdatabase of 600,000 MOFs drawn from open-source repositories. The equations performed consistently well across this expanded dataset, confirming their broad applicability.

Importantly, the study focuses on usable hydrogen capacity, not just total adsorption. This distinction matters because hydrogen that cannot be released under practical conditions is effectively useless for energy applications.

How This Compares to U.S. Energy Targets

The U.S. Department of Energy has set ambitious benchmarks for onboard hydrogen storage systems, including targets of 5.5% hydrogen by weight and 40 grams of hydrogen per liter under real-world conditions. Despite decades of research, no known material has yet achieved both targets simultaneously.

While the new equations do not magically produce a perfect MOF, they provide a powerful tool for identifying the most promising candidates far more quickly than before. Instead of running expensive simulations on millions of hypothetical structures, researchers can now filter candidates almost instantly using a handful of physical parameters.

Why Interpretability Matters

One of the guiding principles behind this work is interpretability. In safety-critical and mission-critical applications—such as energy storage for vehicles or infrastructure—it is not enough for a model to be accurate. Engineers and policymakers need to understand why a prediction is being made.

By matching the performance of black-box models with equations that are easy to interpret and deploy, the study demonstrates that transparency does not have to come at the cost of accuracy.

Broader Implications for Materials Science

Beyond hydrogen storage, this research highlights a broader shift in materials discovery. Physics-informed analytical models offer a bridge between large-scale simulations and real-world design. If similar approaches can be applied to batteries, catalysts, or carbon-capture materials, the pace of clean-energy innovation could accelerate significantly.

Because the equations are computationally inexpensive, they can be embedded directly into automated material-design pipelines, guiding synthesis efforts and reducing wasted experimentation.

A Quick Look at Metal-Organic Frameworks

MOFs are already being explored for applications beyond hydrogen storage, including carbon dioxide capture, gas separation, sensing, and drug delivery. Their modular nature allows scientists to mix and match metal nodes and organic linkers, leading to millions of possible structures. The challenge has never been a lack of options—it has been deciding which ones are worth making.

This new work helps turn that overwhelming design space into something manageable.

Looking Ahead

While there is still work to be done before hydrogen-powered systems become mainstream, tools like these simple equations bring the field closer to practical solutions. By showing that a small number of physical traits can unlock reliable predictions, the study reframes how researchers think about complexity in materials science.

Sometimes, it turns out, simple math really can go a long way.

Research paper: https://journals.aps.org/prxenergy/abstract/10.1103/glnw-drn4