Neural-network-based PDE solvers such as PINNs are hampered by quadrature errors, weakly imposed boundary conditions, and the cost of differentiating the network in space. Finite Element Interpolated Neural Networks (FEINNs) overcome these issues by interpolating the network onto a finite element space. In this talk, we will show how this idea carries over to IGA, leading to IsoGeometric Analysis Quasi-Interpolated Neural Networks (IGAQINNs).

In IGAQINNs, the network is quasi-interpolated onto an IGA spline space and trained by minimising the discrete residual of its quasi-interpolant. We will see that, as a consequence, Dirichlet conditions are imposed strongly, all integrals are computed exactly, and the network never needs to be differentiated in space. Moreover, since the training cost scales with the dimension of the discrete space, the higher accuracy per degree of freedom of IGA compared with FEM directly translates into cheaper training.

We will then present a priori error estimates for the method. The error splits into discretisation, expressivity and optimisation contributions, with constants independent of both the mesh size and the spline degree. Moreover, the network itself retains the optimal convergence order, provided the quasi-interpolant reproduces polynomials of the spline degree.

Finally, we will show numerical results on both forward and inverse problems, such as recovering an unknown diffusion coefficient from partial or noisy observations of the state within a single optimisation loop.