Discrete Solution Operator Learning for Geometry-Dependent PDEs

2026-01-14

AI Summary

This work addresses the challenge that traditional neural operators struggle with discontinuities in discrete structures arising from topological changes, abrupt boundary variations, or shifts in computational domains. To overcome this limitation, the authors propose a novel paradigm termed Discrete Solution Operator Learning (DiSOL), which decomposes the solution process into three learnable stages: local encoding, multiscale assembly, and implicit reconstruction. Unlike conventional approaches that rely on continuous function mappings and smooth geometric variations, DiSOL emphasizes process-level consistency. By integrating classical discretization principles with deep learning, the method encodes local contributions on embedded meshes and aggregates information across multiple scales. It demonstrates high-accuracy and robust predictions—both in-distribution and under strong out-of-distribution scenarios—for problems involving complex geometric changes, including Poisson equations, convection-diffusion, linear elasticity, and heat conduction.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Federated Web and WoT systems, including distributed, federated and edge-based data processing
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• geometry-dependent PDEs
• discrete structural changes
• topological changes
• boundary condition discontinuities
• computational domain variation

Innovation

• Discrete Solution Operator Learning
• geometry-dependent PDEs
• neural operators
• topological changes
• procedural representation