Journal: Nature Communications

Article Title: One-shot learning for solution operators of partial differential equations

doi: 10.1038/s41467-025-63076-z

Figure Lengend Snippet: ( Step 1 ) We select a suitable polygon, such as a rectangle, on a local mesh with step size Δ x 1 and Δ x 2 , and thus define a local domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Omega }$$\end{document} Ω ~ (the black nodes). ( Step 2 ) We select a target mesh node x * and define a local solution operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{{\mathcal{G}}}$$\end{document} G ~ . ( Step 3 ) We learn \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{{\mathcal{G}}}$$\end{document} G ~ using a neural network from a dataset constructed from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{T}}=({f}_{{\mathcal{T}}},{u}_{{\mathcal{T}}})$$\end{document} T = ( f T , u T ) . ( Step 4 ) For a new PDE condition (i.e., a new input function f ), we utilize the pre-trained \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{{\mathcal{G}}}$$\end{document} 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 u 0 ( x ), we apply \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{{\mathcal{G}}}$$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{{\mathcal{G}}}$$\end{document} 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 u 0 ( x ).

Article Snippet: For a new PDE condition f , we choose one of the three approaches to find the global PDE solution using the pre-trained local solution operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{{\mathcal{G}}}$$\end{document} G ~ : fixed-point iteration (FPI), local-solution-operator informed neural network (LOINN) or local-solution-operator informed neural network with correction (cLOINN).

Techniques: Construct