Solving PDEs with manifold learning algorithms

Solving partial differential equations (PDEs) on unknown manifolds has been an important and challenging problem in a large corpus o,f applications of sciences and engineering. The main issue in this computational problem is in the approximation and evaluation of d,ifferential operators and the PDE solution on the unknown manifold given only a finite number of randomly distributed observed point, clouds data. Accurate and robust algorithms for completing these computational tasks require analysis and inference of the underlyi,ng geometrical structure from the available point cloud data that lie on the unknown manifold embedded in a high-dimensional space.,Though the given information is restricted only on point clouds, in practical applications, one is often interested to efficiently e,valuate the PDE solution on the whole manifold, making the problem more difficult. To overcome these issues, we propose to develop P,DE solvers using a manifold learning algorithm. Particularly, the proposed technique leverages the diffusion maps algorithm, which i,s a consistent estimator of second-order elliptic differential operators defined on manifolds. The proposed tasks are to build upon,PI's recent work that modifies the diffusion maps by adding fictitious ghost points to remove the bias near the boundaries and thus,allows for the diffusion maps estimation to be valid even on manifolds with boundaries. The new scheme, the ghost point diffusion ma,ps (GPDM), is effective in solving elliptic PDEs on various types of classical boundary conditions that arise in applications, such,as Dirichlet, Neumann, and Robin boundaries. Based on this encouraging result, we propose to develop: (1) A convergence guarantee so,lver for time-dependent PDEs; (2) A higher-order ghost point extension; (3) A deep learning-based solver for the DM eigenvalue probl,ems; and study (4) the spectral convergence of the numerical solution to the corresponding Laplace-Beltrami eigensolution. Another i,mportant practical consideration is to design an efficient solver such that it is amenable to Bayesian inversion applications to rec,over the posterior distribution of an infinite-dimensional object, such as the diffusion coefficient, from observed PDE solutions co,rrupted by noises. In this application, it is not uncommon that the PDEs need to be solved many times (e.g., more than 105 times) on, the proposed parameters to generate statisticall,chlet boundary conditions, which occur in various applications; and (6) a novel approximation to the intrinsic gradient operator tha,t enables efficient likelihood function evaluations in the MCMC steps. The proposed computational methodological development and the,oretical study are expected to have impacts in several areas of applied mathematics, statistical estimation, and engineering. Partic,ularly, PDEs on manifolds are ubiquitous in the mathematical modeling of physical and engineering applications. In physics, such a p,-dimensional ambient Euclidean domain but have representations on low-dimensional manifolds. Besides scientific and engineering appl,ications, this project is likely to generate new mathematical questions, fostering interactions between computational mathematics an,d data-enabled science. The project will be integrated with a variety of educational activities, such as (1) research training envir,onment for graduate students, involving deep learning, diffusion maps, mathematical analysis, PDE modeling, and scientific computing, and (2) Dissemination through conference presentations, journal publications, software development/data sharing to ensure reproduci