Solving PDEs on Unknown Manifolds with Machine Learning
Senwei Liang, Shixiao W. Jiang, John Harlim, Haizhao Yang
TL;DR
The paper tackles solving elliptic PDEs on unknown manifolds represented by point clouds, introducing a mesh-free framework that couples diffusion maps (DM) and neural networks to approximate the PDE solution. The PDE operator is discretized via DM (including variable-bandwidth DM and ghost-point variants for boundaries), and the solution is learned as a least-squares regression with a neural-network ansatz, enabling a continuous map over the manifold. The authors provide a theoretical decomposition into parametrization and optimization errors, proving convergence for infinite-width/depth networks and for wide two-layer networks under gradient descent, and they validate the approach on multiple synthetic and real-world manifold geometries, showing robust generalization relative to Nyström-based interpolation and FEM benchmarks. The work demonstrates a scalable, mesh-free pathway to PDEs on high- and unknown-co-dimension manifolds, with practical implications for physics-informed learning and geometry-aware simulations. Mathematical formulations such as $(-a+ abla_gig( abla_g uig) ight) = f$ and the DM discretization $L_ ext{ε}$ are central to the method, and the approach yields a continuous NN solution $u(oldsymbol{x}) oughly oldsymbol{x} o u(oldsymbol{x})$ on unseen data points with favorable generalization properties.
Abstract
This paper proposes a mesh-free computational framework and machine learning theory for solving elliptic PDEs on unknown manifolds, identified with point clouds, based on diffusion maps (DM) and deep learning. The PDE solver is formulated as a supervised learning task to solve a least-squares regression problem that imposes an algebraic equation approximating a PDE (and boundary conditions if applicable). This algebraic equation involves a graph-Laplacian type matrix obtained via DM asymptotic expansion, which is a consistent estimator of second-order elliptic differential operators. The resulting numerical method is to solve a highly non-convex empirical risk minimization problem subjected to a solution from a hypothesis space of neural networks (NNs). In a well-posed elliptic PDE setting, when the hypothesis space consists of neural networks with either infinite width or depth, we show that the global minimizer of the empirical loss function is a consistent solution in the limit of large training data. When the hypothesis space is a two-layer neural network, we show that for a sufficiently large width, gradient descent can identify a global minimizer of the empirical loss function. Supporting numerical examples demonstrate the convergence of the solutions, ranging from simple manifolds with low and high co-dimensions, to rough surfaces with and without boundaries. We also show that the proposed NN solver can robustly generalize the PDE solution on new data points with generalization errors that are almost identical to the training errors, superseding a Nystrom-based interpolation method.