Learning Solution Operators for PDEs on Triangular Meshes

Abstract

Partial differential equations are widely used to model physical and geometrical behaviour. Real-time graphics and interactive simulation applications often require fast approximations of these equations. Traditional solvers often require solving large linear systems, which can form a bottleneck for real-time applications. Neural operators aim to learn the direct mapping between input and solution functions. This thesis investigates whether neural operator architectures can learn solution operators for partial differential equations on fixed triangular mesh surfaces. Several model families are compared, including a standard Multilayer Perceptron, DeepONet, a Graph Neural Network, a Multigrid Graph Neural Network, a Spectral Graph Neural Network, and a Hodge Spectral Graph Neural Network. These models are trained and evaluated on three surface-based PDE tasks: the Poisson equation with different scalar right-hand sides, geodesic distance approximation using the heat method with a varying number of source points, and linear elasticity using a Reissner–Mindlin thin shell with different vector-valued force fields. The predictions are evaluated using relative $L_2$ loss, isoline comparisons for scalar fields, error visualisations for displacement fields, and inference speed measurements. The results show that different PDEs favour different architectures. Spectral models perform especially well on the Poisson problem, particularly for smooth right-hand sides. Geodesic distance approximation is harder to learn, although the Hodge Spectral Graph Neural Network performs best overall for this task. The linear elasticity experiments show a different trend: for smoother force fields, the standard Multilayer Perceptron is highly competitive and often achieves the lowest loss, while localised force fields remain more difficult to approximate. This work also shows that neural operators can achieve significant speed-ups over traditional solvers, especially for more complex problems and larger meshes, suggesting a valid use case for these models in real-time applications.