Discrete Solution Operator Learning for Geometry-Dependent PDEs
This addresses geometry-dominated problems in scientific machine learning by providing a complementary paradigm for engineering settings with discrete structural changes.
The paper tackles the problem of solving geometry-dependent PDEs where varying geometry causes discrete structural changes that break smoothness assumptions, by introducing Discrete Solution Operator Learning (DiSOL) to learn discrete solution procedures instead of continuous operators, resulting in stable and accurate predictions across various PDEs under in-distribution and out-of-distribution geometries.
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.