CASIN: Curvature-Aware SImplicial Networks for Time-Dependent PDEs on Unknown Riemannian Manifolds

Partial differential equations (PDEs) are central to mathematical descriptions of physical, biological, and environmental processes, providing a means of representing how systems evolve across multiple interacting variables. In practice, however, observations of such systems are frequently sparse, irregularly distributed, and collected over domains whose geometry is not known in advance. This presents a fundamental challenge in applications such as biomedical imaging, where disease-related patterns develop over curved anatomical surfaces, and environmental monitoring, where observations may be obtained from weather stations distributed unevenly across complex terrain. Although these non-Euclidean settings occur throughout fields ranging from wildfire modeling to healthcare, computational approaches for solving PDEs on unknown geometric domains have received substantially less attention than methods developed for conventional Euclidean spaces. We develop a learning-based framework for approximating the dynamics of time-dependent PDEs directly from observations sampled on unknown Riemannian manifolds. Rather than imposing a predefined coordinate system or requiring explicit knowledge of the underlying geometry, the proposed approach constructs a simplicial representation of the observed domain and uses discrete exterior calculus to approximate the differential operators governing the dynamics. Simplicial convolution provides the principal mechanism for propagating information across this geometric representation, enabling the resulting neural solver to account explicitly for local curvature and higher-order geometric structure. The framework is evaluated on nonlinear reaction–diffusion systems exhibiting complex pattern formation over both synthetic manifolds and real tooth surfaces. In comparisons involving eight state-of-the-art neural PDE solvers across nine experimental configurations, the proposed method produces the smallest error in seven cases. It reduces RMSE substantially relative to the strongest competing method that explicitly incorporates curvature and achieves nearly an order-of-magnitude improvement over approaches that do not account for the underlying geometry. Experiments using real dental scans further show that the proposed approach achieves the best performance for all three reaction–diffusion systems. Finally, we assess its generality on a nonlinear advection equation describing fluid transport on manifolds, demonstrating that the framework extends beyond reaction–diffusion dynamics to other classes of time-dependent PDEs on complex non-Euclidean domains.