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By combining geometry and machine learning, the team has opened the door to solving problems that were previously out of reach, making neural networks not only faster, but smarter in how they learn physics.
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Researchers from the Institute of Cosmos Sciences of the University of Barcelona (ICCUB) have developed a novel machine learning framework that significantly improves the ability to solve complex differential equations, especially in cases where traditional methods struggle. The work, led by Pedro Tarancón-Álvarez and Pablo Tejerina-Pérez, has been published in Communications Physics, a journal from the Nature Portfolio. 
 

Differential equations are essential tools in physics, used to describe phenomena ranging from fluid dynamics to general relativity. However, when these equations become "stiff" (meaning they involve vastly different scales or highly sensitive parameters), they become extremely difficult to solve. This is particularly true for inverse problems, where scientists aim to deduce unknown physical laws or parameters from observed data. 
 

To tackle this challenge, the researchers have enhanced the capabilities of Physics-Informed Neural Networks (PINNs), a type of artificial intelligence that incorporates physical laws into its learning process. Their approach combines two innovative techniques:
 

  • Multi-Head (MH) Training: This allows the neural network to learn a general space of solutions for a family of equations, rather than just one specific case.
  • Unimodular Regularization (UR): Inspired by concepts from differential geometry and general relativity, this technique stabilizes the learning process and improves the network’s ability to generalize to new, more difficult problems.
     

These methods were successfully applied to three increasingly complex systems: the flame equation, the Van der Pol oscillator, and the Einstein Field Equations in a holographic context. In the latter case, the researchers were able to recover unknown physical functions from synthetic data, a task previously considered nearly impossible. 
 

“Recent advances in machine learning training efficiency have made PINNs increasingly popular in the past few years,” explains Pedro Tarancón-Álvarez, PhD candidate at ICCUB. “This framework offers several novel features compared to traditional numerical methods, most notably the ability to solve inverse problems.”

“Solving these inverse problems is like trying to find the solution to a problem that is missing a piece; the correct piece will have a unique solution, incorrect ones may not have a solution, or multiple ones,” adds Pablo Tejerina-Pérez, PhD candidate at ICCUB. “One could try to invent the missing piece of the problem and then see if it can be solved properly – our PINNs do the same, but in a much smarter and efficient way than us.” 
 

The research was carried out in collaboration with Raul Jimenez (ICREA-ICCUB) and Pavlos Protopapas (Harvard University) and was supported by the Spanish Ministry of Science and Innovation and the Maria de Maeztu Excellence Program. 
 

Reference:

Tarancón-Álvarez, P., Tejerina-Pérez, P., Jimenez, R. et al. Efficient PINNs via multi-head unimodular regularization of the solutions space. Commun Phys 8, 335 (2025). https://doi.org/10.1038/s42005-025-02248-1