Physics-Informed Patchwork

We propose Physics-Informed Patchwork (PIP), a domain-decomposition framework intended for a broad class of stationary boundary-value problems through the reuse of the same pretrained boundary-conditioned local neural solver across subdomains. In this study, we assess its feasibility for the Laplace equation with Dirichlet boundary conditions. The general idea is to train the neural model to approximate solutions of a stationary partial differential equation (PDE) on the unit square, with boundary profiles provided as inputs. A global solution is then assembled using the model with frozen weights by optimizing the internal interface profiles between adjacent subdomains so as to reduce mismatches in solution values and normal derivatives across their interfaces. We demonstrate the method using a fully connected neural network as the local solver. In local-solver tests, the mean relative error is 2.064% on the in-distribution (ID) dataset and 6.853–6.954% on two out-of-distribution (OOD) datasets. For the tested global boundary-value problems, computational scaling is investigated up to 104 subdomains. The measured computational time of global-domain assembly exhibits near-linear scaling with the number of subdomains. For the tested nonrectangular global domains assembled from square subdomains, the mean relative error with respect to finite-difference reference solutions is 0.44–1.14%, while for problems with analytical harmonic solutions it is 0.4–1.3%. For the stationary Laplace equation considered here, these results demonstrate the feasibility of assembling global solutions on large and geometrically nontrivial domains by repeatedly reusing a single local neural model without retraining its weights for each new global problem.