Scientific machine learning (SciML) explores the development of neural network models to approximate solutions to Partial Differential Equations (PDEs). However, there exists a significant research gap when computational domains are arbitrary manifolds, which are common in real-world scientific and engineering applications. The inherent challenge arises when calculating differential operators defined on curved surfaces, particularly in scenarios where surface parameterization is unavailable. In this paper, we present a neural network-based method for solving PDEs on surfaces described only by point clouds, without any other geometrical priors. Our method comprises two steps—local surface approximation based on graph neural networks and solving PDEs on point clouds. For surface reconstruction, our graph neural networks can be generalized based on the predictions of simple geometries during training to significantly more complicated surfaces for evaluation. The proposed approach demonstrates its capacity to learn geometric features from point cloud data without requiring external datasets, offers superior performance compared to benchmark models across various PDE types, and exhibits robustness in handling complex surfaces, non-uniform point distributions, and noise.

The modeling and control of complex physical systems are essential in real-world problems. We propose a novel framework that is generally applicable to solving PDE-constrained optimal control problems by introducing surrogate models for PDE solution operators with special regularizers. The procedure of the proposed framework is divided into two phases: solution operator learning for PDE constraints (Phase 1) and searching for optimal control (Phase 2). Once the surrogate model is trained in Phase 1, the optimal control can be inferred in Phase 2 without intensive computations. Our framework can be applied to both data-driven and data-free cases. We demonstrate the successful application of our method to various optimal control problems for different control variables with diverse PDE constraints from the Poisson equation to Burgers' equation.