Partial differential equations (PDEs) are a class of mathematical problems that represent the interplay of multiple variables, and therefore have predictive power when it comes to complex physical systems. Solving these equations is a perpetual challenge, however, and current computational techniques for doing so are time-consuming and expensive.
Now, research from the University of Utah’s John and Marcia Price College of Engineering is showing a way to speed up this process: encoding those equations in light and feeding them into their newly designed “optical neural engine,” or ONE.
The researchers’ ONE combines diffractive optical neural networks and optical matrix multipliers. Rather than representing PDEs digitally, the researchers represented them optically, with variables represented by the various properties of a light wave, such as its intensity and phase. As a wave passes through the ONE’s series of optical components, those properties gradually shift and change, until they ultimately represent the solution to the given PDE.
The research was led by Weilu Gao, assistant professor in the Department of Electrical & Computer Engineering, and Ruiyang Chen, a Ph.D. candidate in Gao’s research group. They published a study demonstrating this optical neural engine in the journal Nature Communications.
“Partial differential equations are a powerful computational tool to simulate physics problems instead of performing expensive and time-consuming real-world experiments,” Chen said. “However, the current numerical simulation method is slow and requires a lot of computing resources, and even electronic machine learning techniques are not fast enough.”
The machine learning techniques and digital neural networks currently used to solve PDEs involve passing the equation through a network of computational nodes, each of which weights its output as it passes it on to the next node. As the signal makes its way through the network, the correct solution becomes weighted most heavily and ends up as the output.
The researchers’ ONE takes this concept and applies it to photonic devices.
“The ONE takes the spatiotemporal data of an input physical quantity, which is a function of positions and time, to predict the spatiotemporal data of an output physical quantity as a function of positions and time,” Gao said.
Electronic machine learning techniques can produce a similar output, but at slower speeds and higher energy costs.
“This optical approach accelerates the machine learning process and requires less energy as compared to an electronic approach,” said lead author and former Gao lab member Yingheng Tang, now a research scientist with Lawrence Berkeley National Laboratory.
The researchers demonstrated their ONE’s capabilities on a number of PDEs, including the Darcy flow equation, the magnetostatic Poisson’s equation in demagnetization, and the Navier-Stokes equation in incompressible fluid.
“The Darcy flow equation, for example, describes a fluid flow through a porous medium,” Gao said. “Given data about the permeability and pressure fields inside a given medium, the ONE architecture essentially learns the mapping between those qualities, and can predict flow properties without having to experiment.”
“This research offers a versatile and powerful platform for large-scale scientific and engineering computation, such as geology and chip design,” Gao said.
More information:
Yingheng Tang et al, Optical neural engine for solving scientific partial differential equations, Nature Communications (2025). DOI: 10.1038/s41467-025-59847-3
University of Utah
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‘Optical neural engine’ can solve partial differential equations (2025, June 10)
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