ParaFIND: Parameter Field Inference on Non-uniform Domains using Neural Network

Real-world physical phenomena are often governed by Partial Differential Equations (PDEs) influenced by spatially distributed properties/parameters. While many deep learning techniques offer reasonable parameter estimation results for simple cases, they struggle to generalize to irregular geometries, limiting their applicability to a narrow range of problems. To address this limitation, we propose ParaFIND, a novel approach for estimating unknown PDE parameter fields distributed on non-uniform domains from scarce observations of the system's response. Our method leverages the Finite Element Method (FEM) for space discretization and learns parameters modeled as functions of space from their mesh representation. This innovative approach enables accurate parameter estimation even in complex geometrical settings. We demonstrate the robustness of our model under limited sparse data constraints using as few as 108 data samples. Our numerical simulations validate the effectiveness of ParaFIND in handling various irregularities in a given geometry, showcasing its potential for broader applications in real-world scenarios.