Energy minimisation in nonlinear function spaces for the approximation of elliptic PDEs

This thesis investigates nonlinear methods for approximating solutions to elliptic partial differential equations via energy minimisation. The first part introduces adaptive quadratures for physics-informed neural networks, leveraging piecewise-linear approximations of activation functions to enable accurate integration with few evaluation points. The second part presents a general optimisation framework for nonlinear approximation spaces, establishing convergence under broad structural conditions. The framework is realised using free-knot B-splines, where knot positions adapt to the local features of the solution. Numerical experiments reveal marked improvements in accuracy compared to conventional linear methods, highlighting the potential and challenges of nonlinear discretisations in PDE solvers.