In this paper, we develop a class of interacting particle Langevin algorithms to solve inverse problems
for partial differential equations (PDEs). In particular, we leverage the statistical finite elements (statFEM)
formulation to obtain a finite-dimensional latent variable statistical model where the parameter is that
of the (discretised) forward map and the latent variable is the statFEM solution of the PDE which is
assumed to be partially observed. We then adapt a recently proposed expectation-maximisation like
scheme, interacting particle Langevin algorithm (IPLA), for this problem and obtain a joint estimation
procedure for the parameters and the latent variables. We consider three main examples: (i) estimating the
forcing for linear Poisson PDE, (ii) estimating the forcing for nonlinear Poisson PDE, and (iii) estimating
diffusivity for linear Poisson PDE. We provide computational complexity estimates for forcing estimation
in the linear case. We also provide comprehensive numerical experiments and preconditioning strategies
that significantly improve the performance, showing that the proposed class of methods can be the choice
for parameter inference in PDE models.