Applied Math Colloquium: Gianluca Fabiani

title:

Random Projection Neural Networks for Function and Operator Approximation with Applications to Nonlinear PDEs

abstract:

How far can scientific machine learning methods go in solving, learning, and analysing nonlinear differential equations, and at what computational cost compared with classical numerical analysis? This talk addresses this question by following a research line motivated by the structural and computational limitations of fully trained deep neural networks in scientific computing.

Training deep neural networks requires solving high-dimensional non-convex optimisation problems with no general convergence guarantees, and is often costly in stiff or multiscale regimes. This motivates a simplification in which the nonlinear feature map is fixed a priori through random sampling, reducing training to a linear least-squares problem with favourable conditioning. This is the core idea behind Random Projection Neural Networks and related random feature methods, whose analysis connects the Johnson–Lindenstrauss lemma, universal approximation results for random features, and geometrically informed sampling strategies. In this context, Random Projection Neural Networks of best approximation are introduced, which recover classical approximation properties in the neural setting, including polynomial best approximation and exponential convergence for analytic target functions, clarifying the link with classical approximation theory.

Within this framework, physics-informed random feature models are compared with classical physics-informed neural networks and standard discretisation schemes, showing competitive accuracy and reduced computational cost in stiff ordinary differential equations, index-1 differential-algebraic equations, and steady nonlinear partial differential equations. In addition to forward problems, the same linear-algebraic structure can be exploited for stability and bifurcation analysis, where properties of the resulting collocation systems motivate matrix-free Krylov–Arnoldi methods for eigenvalue computation and continuation of nonlinear PDE equilibria.

For operator approximation, I will discuss RandONets, shallow neural architectures with random projections designed for learning solution operators of parametric differential equations. These models provide efficient surrogates for nonlinear maps between function spaces and often outperform deeper operator learning architectures, such as DeepONets, in terms of training cost while retaining comparable accuracy on standard benchmarks.

More information: https://gianlucafabiani.github.io/