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Here are some demos of solving inverse problems of PDEs.

Inverse problem for the Poisson equation with unknown forcing field with training points resampling

Inverse problem for Brinkman-Forchheimer model

Inverse problem for the diffusion-reaction system with a space-dependent reaction rate

Inverse problem for the Navier-Stokes equation of incompressible flow around cylinder

Inverse problem for the fractional Poisson equation in 1D

Inverse problem for the fractional Poisson equation in 2D