A research team from the Skoltech AI Center proposed a new neural network architecture for generating structured curved coordinate grids, an important tool for calculations in physics, biology, and even finance. The study is published in the Scientific Reports journal.
“Building a coordinate grid is a key task for modeling. Breaking down a complex space into manageable pieces is necessary, as it allows you to accurately determine the changes in different quantities—temperature, speed, pressure, and so on,” commented the lead author of the paper, Bari Khairullin, a Ph.D. student from the Computational and Data Science and Engineering program at Skoltech.
“Without a good grid, calculations become either inaccurate or impossible. In physics, they help model the movement of liquids and gases, in biology, tissue growth and drug distribution, and in finance, they predict market fluctuations. The proposed approach opens up new possibilities in building grids using artificial intelligence.”
Traditional approaches, such as solving Winslow equations, rely on numerical solutions of partial differential equations and do not provide exact analytic expressions for the Jacobian of the transformation.
In contrast, the proposed architecture treats the neural network as a diffeomorphism between the computational and physical domains, enabling exact Jacobian evaluation and fast mesh refinement via a single forward pass.
The team considered two approaches—with physics-informed loss terms (Physics-Informed Neural Networks, PINN) and without them. In the latter case, the authors derive analytic formulas that link the weights of the network to the non-degeneracy of the mapping. These estimates allow for control over the sign and lower bound of the Jacobian determinant, ensuring bijectivity, and preventing mesh folding.
A key difference from the earlier MGNet architecture lies in the use of residual connections between all layers. This design models the transformation as a sequence of small deformations, starting from the identity map and allowing for localized correction and better control over regularity.
The experiments show that the PINN-based method is capable of generating high-quality grids even on multiple connected domains. Numerical results confirm the method’s potential in applications where accurate geometry representation is critical for solving partial differential equations.
“Processing geometric transformations using neural networks can become a new stage in the development of grid generation methods,” explains study co-author Sergey Rykovanov, the head of the Artificial Intelligence and Supercomputing Laboratory at the Skoltech AI Center. “The next step will be to generalize the results to 3D areas.”
Some computations were performed on the Zhores supercomputer at Skoltech.
More information:
Bari Khairullin et al, Neural networks for structured grid generation, Scientific Reports (2025). DOI: 10.1038/s41598-025-97059-3
Skolkovo Institute of Science and Technology
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AI automates structured grid generation for better simulations (2025, April 28)
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