$Δ$-PINNs: physics-informed neural networks on complex geometries

TL;DR

Δ-PINNs address the limitation of conventional PINNs in handling complex geometries by encoding domain geometry with Laplace-Beltrami eigenfunctions, turning intrinsic geometry into neural network inputs. The method replaces Cartesian coordinates with a finite set of eigenfunctions, enabling FE-based computation of PDE operators on manifolds and allowing mini-batch training on large meshes. Across Eikonal, heat transfer, Poisson, hyperelasticity, and geodesic-distance problems on coils, heat sinks, and bunny surfaces, Δ-PINNs consistently outperform standard PINNs and show competitive performance with graph-based methods, while offering scalable eigenfunction precomputation. This approach broadens PINN applicability to realistic geometric domains and paves the way for extensions to 3D solids and time-dependent problems, with the potential for mesh-free variants in the future.

Abstract

Physics-informed neural networks (PINNs) have demonstrated promise in solving forward and inverse problems involving partial differential equations. Despite recent progress on expanding the class of problems that can be tackled by PINNs, most of existing use-cases involve simple geometric domains. To date, there is no clear way to inform PINNs about the topology of the domain where the problem is being solved. In this work, we propose a novel positional encoding mechanism for PINNs based on the eigenfunctions of the Laplace-Beltrami operator. This technique allows to create an input space for the neural network that represents the geometry of a given object. We approximate the eigenfunctions as well as the operators involved in the partial differential equations with finite elements. We extensively test and compare the proposed methodology against traditional PINNs in complex shapes, such as a coil, a heat sink and a bunny, with different physics, such as the Eikonal equation and heat transfer. We also study the sensitivity of our method to the number of eigenfunctions used, as well as the discretization used for the eigenfunctions and the underlying operators. Our results show excellent agreement with the ground truth data in cases where traditional PINNs fail to produce a meaningful solution. We envision this new technique will expand the effectiveness of PINNs to more realistic applications.

Figures (14)

• Figure 1: Learning the Eikonal equation on a coil. Top row: the 1st, 10th, 50th and 100th Laplace-Beltrami eigenfunction of the geometry. Bottom row, first: the ground truth solution of the Eikonal equation, which represents the geodesic distance, and the data points used for training shown as gray spheres. Second, the solution of $\Delta$-PINNs, our proposed method trained with 50 eigenfunctions. Third, the traditional PINNs approximation. Last, the approximate solution of a physics-informed graph-convolution network.
• Figure 2: Traditional PINNs vs. $\Delta$-PINNs architecture. In traditional PINNs, the input layer is the vector of coordinates. With $\Delta$-PINN, the input layer is the value of the eigenfunctions at the coordinate of interest. In this way we can encode much more information about the geometry.
• Figure 3: Accuracy of the coil example with the Eikonal equation. Correlation between predicted and ground truth values of geodesic distance for $\Delta$-PINNs (first), $\Delta$-PINNs trained without the loss functions that includes the Eikonal equation (second), traditional PINNs (third) and a graph convolutional network (last).
• Figure 4: Learning the temperature distribution of a heat sink from boundary measurements. First, the boundary conditions and the finite element mesh used to create the solution. Second, the ground truth, and in the learned solution of (third) $\Delta$-PINNs, (forth) signed distance function PINNs and (last) traditional PINNs. The data points are shown on the ground truth panel in black.
• Figure 5: Accuracy the heat sink example. Correlation between predicted and ground truth values of temperature for $\Delta$-PINNs (left), signed distance function PINNs (center) and traditional PINNs (right).