  In this paper, we evaluate the effectiveness of deep operator networks
(DeepONets) in solving both forward and inverse problems of partial
differential equations (PDEs) on unknown manifolds. By unknown manifolds, we
identify the manifold by a set of randomly sampled data point clouds that are
assumed to lie on or close to the manifold. When the loss function incorporates
the physics, resulting in the so-called physics-informed DeepONets
(PI-DeepONets), we approximate the differentiation terms in the PDE by an
appropriate operator approximation scheme. For the second-order elliptic PDE
with a nontrivial diffusion coefficient, we approximate the differentiation
term with one of these methods: the Diffusion Maps (DM), the Radial Basis
Functions (RBF), and the Generalized Moving Least Squares (GMLS) methods. For
the GMLS approximation, which is more flexible for problems with boundary
conditions, we derive the theoretical error bound induced by the approximate
differentiation. Numerically, we found that DeepONet is accurate for various
types of diffusion coefficients, including linear, exponential, piecewise
linear, and quadratic functions, for linear and semi-linear PDEs with/without
boundaries. When the number of observations is small, PI-DeepONet trained with
sufficiently large samples of PDE constraints produces more accurate
approximations than DeepONet. For the inverse problem, we incorporate
PI-DeepONet in a Bayesian Markov Chain Monte Carlo (MCMC) framework to estimate
the diffusion coefficient from noisy solutions of the PDEs measured at a finite
number of point cloud data. Numerically, we found that PI-DeepONet provides
accurate approximations comparable to those obtained by a more expensive method
that directly solves the PDE on the proposed diffusion coefficient in each MCMC
iteration.
