Neural Green's Functions

TL;DR

Neural Green's Function, a neural solution operator for linear partial differential equations (PDEs) whose differential operators admit eigendecompositions, is introduced, a neural solution operator for linear PDEs whose differential operators admit eigendecompositions inspired by Green's functions.

Abstract

We introduce Neural Green's Function, a neural solution operator for linear partial differential equations (PDEs) whose differential operators admit eigendecompositions. Inspired by Green's functions, the solution operators of linear PDEs that depend exclusively on the domain geometry, we design Neural Green's Function to imitate their behavior, achieving superior generalization across diverse irregular geometries and source and boundary functions. Specifically, Neural Green's Function extracts per-point features from a volumetric point cloud representing the problem domain and uses them to predict a decomposition of the solution operator, which is subsequently applied to evaluate solutions via numerical integration. Unlike recent learning-based solution operators, which often struggle to generalize to unseen source or boundary functions, our framework is, by design, agnostic to the specific functions used during training, enabling robust and efficient generalization. In the steady-state thermal analysis of mechanical part geometries from the MCB dataset, Neural Green's Function outperforms state-of-the-art neural operators, achieving an average error reduction of 13.9\% across five shape categories, while being up to 350 times faster than a numerical solver that requires computationally expensive meshing.

Figures (5)

• Figure 1: Method overview. Given query points representing the geometry of the problem domain where the solution function is to be predicted, Neural Green’s Function extracts neural features that are subsequently used to approximate the Green’s function and to predict the differential quantities required to evaluate Eqn. \ref{['eqn:system_solution']}. The example is rendered using a tetrahedral mesh, which is used solely for visualization purposes.
• Figure 2: Example shapes from our dataset. Our dataset comprises diverse mechanical part shapes from the MCB dataset Kim:2020MCB, designed to evaluate the generalizability of learning-based PDE solvers across shape variations.
• Figure 3: Qualitative comparison. For each ground-truth solution in columns one and six, the per-point $L_2$ errors from various methods are visualized in the error maps across the remaining columns. Red denotes higher values in the columns labeled "GT", while in the error maps, brighter colors indicate greater errors. Neural Green’s Function demonstrates its ability to accurately predict solutions on irregular geometries from diverse domains. Compared to state-of-the-art neural operators trained to directly map source and boundary functions to solutions, our method achieves lower errors, as indicated by the darker regions in the error maps. The examples are rendered using tetrahedral meshes, that are used solely for visualization purposes. Best viewed when zoomed-in.
• Figure 4: Qualitative results from the ablation study. For each ground-truth solution in columns one and four, we visualize the per-point $L_2$ errors from variants of our method in the remaining columns. Red denotes higher values in the columns labeled "GT", while in the error maps, brighter colors indicate greater errors. The examples are rendered using tetrahedral meshes, that are used solely for visualization purposes. Best viewed when zoomed-in.
• Figure 5: Additional qualitative comparison. For each ground-truth solution in columns one and six, the per-point $L_2$ errors from various methods are visualized in the error maps across the remaining columns.