Learning to Solve PDE-constrained Inverse Problems with Graph Networks
This addresses inverse problems in science and engineering, offering a faster and more accurate method for tasks like wave or Navier-Stokes equations, but it is incremental as it builds on existing GNN techniques.
The paper tackles PDE-constrained inverse problems, such as recovering initial conditions or parameters from sparse measurements, using graph neural networks (GNNs) with autodecoder-style priors, achieving more accurate estimates than other learned approaches and computational speedups of up to 90x compared to principled solvers.
Learned graph neural networks (GNNs) have recently been established as fast and accurate alternatives for principled solvers in simulating the dynamics of physical systems. In many application domains across science and engineering, however, we are not only interested in a forward simulation but also in solving inverse problems with constraints defined by a partial differential equation (PDE). Here we explore GNNs to solve such PDE-constrained inverse problems. Given a sparse set of measurements, we are interested in recovering the initial condition or parameters of the PDE. We demonstrate that GNNs combined with autodecoder-style priors are well-suited for these tasks, achieving more accurate estimates of initial conditions or physical parameters than other learned approaches when applied to the wave equation or Navier-Stokes equations. We also demonstrate computational speedups of up to 90x using GNNs compared to principled solvers. Project page: https://cyanzhao42.github.io/LearnInverseProblem