Solving forward and inverse PDE problems on unknown manifolds via physics-informed neural operators

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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.