There is a particular bridge that inspires Rafael Pastrana, a Princeton University Ph.D candidate in architecture. The bridge awes Pastrana for its strong geometry and mechanical design. So, is it a beautiful medieval arch bridge in the tradition of Italian architecture? An ancient South American wonder? Or perhaps one of those expansive bridges of the Golden Coast frequented by tourists for their astonishing views?

No, it’s the Williams Crossing Pedestrian Bridge in Tulsa, Oklahoma. Constructed of weathered steel, spanning over the Arkansas River, and officially opened in Sept. 2024. Most notable, however, is the bridge’s extreme, efficient use of material. The steel plates that construct the bridge are just three inches thick. When divided by the span length, the bridge arches have an unbraced span to thickness ratio of just 1 to 43. Compare that to the span to thickness of an eggshell, which is 1 to 50, and it becomes apparent the incredible thinness of material in relation to how much weight the bridge can hold.

“What I really like about giving the Oklahoma bridge as an example is that it's unexpectedly beautiful,” said Pastrana. Even more than its aesthetics, the bridge is important because it’s a display of efficient material use in long-span structures built in the United States within the last few years. Pastrana wants to demonstrate it’s possible to build structures that use material judiciously anywhere, even where you’d least expect it. 

“My interest during grad school has been in computing efficient geometry for long span structures, like bridges, towers, and roofing systems,” said Pastrana, who is pursuing a graduate certificate in Statistics and Machine Learning alongside his Ph.D. By developing new numerical methods, Pastrana is working on producing optimized shapes that allow for an architectural structure to use as little material as possible in order to reduce their cost and environmental footprint, while also still functioning in the way that it needs to and being visually appealing on top of it all.

“A structure should not only be safe and structurally efficient, but it also has to work architecturally and has to be buildable,” said Pastrana. With so many constraints to satisfy, the design space of possible architectural structures is massive. “In order to search that space efficiently and automatically, we're using machine learning tools.”

The secret sauce

In the architecture program at Princeton, Pastrana is on the computation and energy track. He started out getting his undergraduate degree in civil engineering at Tec de Monterrey in Mexico before going on to work at a structural engineering firm in Germany. It was there at the firm that Pastrana gained his first experience using computation to optimize efficient structures. He continued to expand upon his experience with computation while earning a master’s degree in architecture at ETH Zurich. 

He began pursuing a Ph.D. at Princeton “due to this passion to make computational tools for shape making that could help us build more structures like the Oklahoma bridge – elegant, efficient, and environmentally responsible.”

Now, Pastrana is at work developing and exploring his own methods of computing structural shapes that are both functional and judicious in the use of materials. “One of the secret sauces powering the methods I work with to create good shapes is machine learning,” said Pastrana. 

Pastrana uses differentiable physics models to find shapes and designs that would follow the constraints needed for an efficient, usable structure. However, the computation process tends to be lengthy, with the model often taking significant time to generate design options. “Ideally, we want to accelerate that process so that you can quickly iterate over multiple possible designs,” said Pastrana. Real-world architecture and engineering projects require strict timeliness and adherence to deadlines. “We're using machine learning models to accelerate the search process.” 

In addition to timeliness, assurance of mechanical integrity is essential. “People’s safety could be at stake, and traditional machine learning models do not offer the guarantees we need,” said Pastrana. The answer? “We developed new hybrid models that combine neural networks with physics models to get the best of both worlds.”

Teaming up with researchers in the lab of Ryan Adams, professor of computer science, Pastrana has successfully coupled neural networks and physical models to generate feasible design shapes in real-time.“Neural networks are great at generating images and eliciting text,” said Pastrana. “But, when connected with a physics model, they can also excel at producing physically valid design solutions in milliseconds.”

In particular, Pastrana produces efficient shapes adhering to funicular geometry, meaning the shapes bear loads mainly through the forces of axial compression or tension. A chain that hangs under its own weight is a quintessential example of funicular geometry. According to funicular geometry, if you were to flip the chain, it would become an arch that bears loads mainly through axial compression. This efficient mechanical behavior reduces material consumption.

Pastrana validates the functionality of the shape produced by the coupled neural network and physical simulation by building a prototype. And it worked. Because the shape stands under compression, the bricks used to construct the prototype were not mechanically attached but instead could stand just under the weight of being side-by-side in the arrangement of the shape for the design. 

“Rafael is not only really good at machine learning and structural engineering, he can actually build things, he can design,” said Sigrid Adriaenssens, professor of civil and environmental engineering and Pastrana’s advisor. “He is multi-talented.”

Adriaenssens emphasizes that the shapes optimized in Pastrana’s work are exciting because they’re both visually appealing while also taking into account how a structure will behave. “In Rafael's models, we capture the mechanical behavior accurately, all the structures factor in gravity and material characteristics, many of the things needed for practical structural design are included,” said Adriaenssens.

Many modern structures can be bulky and inefficiently shaped, but overall easier and cheaper to construct. Lightweight structures save money on material but may be harder to build than their bulky counterparts. For that reason, architectural engineers like Pastrana are looking for innovative ways to improve how efficient structures are constructed. “There's an opportunity to improve how we build these things so that we don't have to compromise on carbon efficiency or fabrication innovation,” said Pastrana. “We can and should design better structures.”

The use of neural networks in accelerating the optimization of structural shapes is just one way AI technology can be used to improve the process from design to final structure. “I would like to see a new wave of products for structural engineers that leverage some of this AI technology to automate time-consuming, repetitive tasks in the structural engineering process,” said Pastrana. “These tools will empower designers to create more efficient architectural structures.”