Fast simulations on domains of changing shape

The paper Reduced order modeling of partial differential equations on parameter-dependent domains using deep neural networks (M. Bukač, I. Mikuš, B. Muha, D. Vlah) has been published in Computers & Mathematics with Applications (vol. 216, 2026, pp. 282–300). It is available at doi.org/10.1016/j.camwa.2026.05.008, with a preprint at arXiv:2407.17171.

Partial differential equations often depend on parameters and must be solved many times. Reduced order models learn from precomputed solutions and then quickly approximate the solution for new parameters. When the domain itself changes, for example the shape of a blood vessel or the position of an obstacle in a channel, a good description of the domain is needed, which is hard when the shape is complex or changes topology.

The authors propose a framework in which deep neural networks extract the domain description from data, without user-defined control points and without a finite-element mesh, a common situation in biomedical applications. Both the solution and the domain are thus compressed into a low-dimensional latent representation.

The research was partly funded by the VALOR project.