Fig. 1: Workflow of the one-shot learning method for solution operators.
From: One-shot learning for solution operators of partial differential equations
(Step 1) We select a suitable polygon, such as a rectangle, on a local mesh with step size Δx1 and Δx2, and thus define a local domain \(\tilde{\Omega }\) (the black nodes). (Step 2) We select a target mesh node x* and define a local solution operator \(\tilde{{\mathcal{G}}}\). (Step 3) We learn \(\tilde{{\mathcal{G}}}\) using a neural network from a dataset constructed from \({\mathcal{T}}=({f}_{{\mathcal{T}}},{u}_{{\mathcal{T}}})\). (Step 4) For a new PDE condition (i.e., a new input function f), we utilize the pre-trained \(\tilde{{\mathcal{G}}}\) to find the corresponding PDE solution by using one of the following approaches. (Approach 1, FPI) We consider points on an equispaced global mesh. Starting with an initial guess u0(x), we apply \(\tilde{{\mathcal{G}}}\) iteratively to update the PDE solution until it is converged. (Approach 2, LOINN) We use a network to approximate the PDE solution. We apply \(\tilde{{\mathcal{G}}}\) at different random locations to compute the loss function. (Approach 3, cLOINN) We use a network to approximate the difference between the PDE solution and the given u0(x).
