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.

翻译：本文评估了深度算子网络（DeepONets）在未知流形上求解偏微分方程（PDEs）正反问题的有效性。所谓未知流形，是指通过一组随机采样的数据点云来识别的流形，这些点云被假定位于流形上或其附近。当损失函数融入物理信息，形成所谓的物理信息深度算子网络（PI-DeepONets）时，我们通过适当的算子逼近方案来近似PDE中的微分项。对于具有非平凡扩散系数的二阶椭圆型PDE，我们采用以下方法之一来逼近微分项：扩散映射（DM）、径向基函数（RBF）以及广义移动最小二乘法（GMLS）。针对更适用于带边界条件问题的GMLS逼近，我们推导了由近似微分所引致的理论误差界。数值实验表明，对于带/不带边界的线性和半线性PDE，DeepONet在多种扩散系数类型（包括线性、指数、分段线性和二次函数）下均能获得精确解。当观测数据量较少时，使用足够大量PDE约束样本训练的PI-DeepONet能比DeepONet产生更精确的逼近解。对于反问题，我们将PI-DeepONet嵌入贝叶斯马尔可夫链蒙特卡洛（MCMC）框架中，通过有限点云数据测得的含噪PDE解来估计扩散系数。数值结果表明，PI-DeepONet提供的逼近精度与一种更为昂贵的方法相当，后者需要在每次MCMC迭代中直接对提议的扩散系数求解PDE。