Methods to Recover Unknown Processes in Partial Differential Equations Using Data
This work addresses a domain-specific problem in computational mathematics for researchers and practitioners dealing with PDE modeling and data-driven recovery, but it appears incremental as it builds on existing numerical approaches.
The paper tackles the problem of identifying unknown processes in time-dependent partial differential equations (PDEs) using observational data, with applications to advection-diffusion type PDEs, and presents numerical methods like Galerkin and collocation algorithms, demonstrating their performance through various examples.
We study the problem of identifying unknown processes embedded in time-dependent partial differential equation (PDE) using observational data, with an application to advection-diffusion type PDE. We first conduct theoretical analysis and derive conditions to ensure the solvability of the problem. We then present a set of numerical approaches, including Galerkin type algorithm and collocation type algorithm. Analysis of the algorithms are presented, along with their implementation detail. The Galerkin algorithm is more suitable for practical situations, particularly those with noisy data, as it avoids using derivative/gradient data. Various numerical examples are then presented to demonstrate the performance and properties of the numerical methods.