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Neural networks meet isogeometric analysis

Quasi-interpolating neural networks onto spline spaces to combine the flexibility of neural PDE solvers with the accuracy and structure of IGA.

Role: First author
With: C. Giannelli, C. Marcati, G. Sangalli, A. Reali
Tools: Python, JAX
Status: Paper in preparation

The problem

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.

Approach

IsoGeometric Analysis Quasi-Interpolated Neural Networks (IGAQINNs) carry this idea over to isogeometric analysis. The network is quasi-interpolated onto an IGA spline space and trained by minimising the discrete residual of its quasi-interpolant. As a consequence:

Dirichlet conditions are imposed strongly;
all integrals are computed exactly;
the network never needs to be differentiated in space;
since the training cost scales with the dimension of the discrete space, the higher accuracy per degree of freedom of IGA compared with FEM translates directly into cheaper training.

Analysis and results

A priori error estimates split the error into discretisation, expressivity and optimisation contributions, with constants independent of both the mesh size and the spline degree; the network itself retains the optimal convergence order, provided the quasi-interpolant reproduces polynomials of the spline degree. Numerical results cover forward and inverse problems — for example, recovering an unknown diffusion coefficient from partial or noisy observations of the state within a single optimisation loop.

Early results were presented at HOFEIM 2026 (Siena) and in a seminar at the Università di Firenze.