Discrete Solution Operator Learning for Geometry-Dependent PDEs

Neural operator learning accelerates PDE solution by approximating operators as mappings between continuous function spaces. Yet in many engineering settings, varying geometry induces discrete structural changes, including topological changes, abrupt changes in boundary conditions or boundary types, and changes in the computational domain, which break the smooth-variation premise. Here we introduce Discrete Solution Operator Learning (DiSOL), a complementary paradigm that learns discrete solution procedures rather than continuous function-space operators. DiSOL factorizes the solver into learnable stages that mirror classical discretizations: local contribution encoding, multiscale assembly, and implicit solution reconstruction on an embedded grid, thereby preserving procedure-level consistency while adapting to geometry-dependent discrete structures. Across geometry-dependent Poisson, advection-diffusion, linear elasticity, as well as spatiotemporal heat conduction problems, DiSOL produces stable and accurate predictions under both in-distribution and strongly out-of-distribution geometries, including discontinuous boundaries and topological changes. These results highlight the need for procedural operator representations in geometry-dominated problems and position discrete solution operator learning as a distinct, complementary direction in scientific machine learning.

Finite-PINN: A Physics-Informed Neural Network Architecture for Solving Solid Mechanics Problems with General Geometries

PINN models have demonstrated impressive capabilities in addressing fluid PDE problems, and their potential in solid mechanics is beginning to emerge. This study identifies two key challenges when using PINN to solve general solid mechanics problems. These challenges become evident when comparing the limitations of PINN with the well-established numerical methods commonly used in solid mechanics, such as the finite element method (FEM). Specifically: a) PINN models generate solutions over an infinite domain, which conflicts with the finite boundaries typical of most solid structures; and b) the solution space utilised by PINN is Euclidean, which is inadequate for addressing the complex geometries often present in solid structures. This work proposes a PINN architecture used for general solid mechanics problems, termed the Finite-PINN model. The proposed model aims to effectively address these two challenges while preserving as much of the original implementation of PINN as possible. The unique architecture of the Finite-PINN model addresses these challenges by separating the approximation of stress and displacement fields, and by transforming the solution space from the traditional Euclidean space to a Euclidean-topological joint space. Several case studies presented in this paper demonstrate that the Finite-PINN model provides satisfactory results for a variety of problem types, including both forward and inverse problems, in both 2D and 3D contexts. The developed Finite-PINN model offers a promising tool for addressing general solid mechanics problems, particularly those not yet well-explored in current research.