DeepONet Surrogate Modeling for Continuum Mechanics in Unbounded Domains: Bypassing BEM Scalability via Operator Learning

Resumo

Continuum mechanics problems over unbounded domains demand the use of mathematical models due to the nature of these systems being governed by partial differential equations. One such tool is the Boundary Element Method (BEM), which reformulates a given boundary-value problem into equivalent integral equations posed solely on the domain’s boundary. However, classical numerical approaches such as BEM involve computing a large number of numerical integrals, representing a significant computational cost, especially in large-scale problems, as the total computing time for this method scales poorly with problem size due to dense kernel evaluations. This work presents a data-driven approach using DeepONet architectures trained on synthetic data from analytical solutions and numerical simulations. The model employs a branch-trunk structure, with the branch encoding system parameters and the trunk evaluating field points in defined domains. After training, it performs fixed-cost inference independent of problem scale, enabling the evaluation of solutions in milliseconds. We demonstrate close correspondence with analytical solutions for the Kelvin problem in elasticity, as well as problems involving harmonic surface loads for linear-elastic, isotropic material bodies in a half-space.