Gianluca Fabiani

Gianluca Fabiani

Postdoctoral Fellow at the Hopkins Extreme Materials Institute (HEMI)

Preprints

You can also find my preprints on my Google Scholar profile.

Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approach

Online in Arxiv, 2026

We present a matrix-free Krylov–Arnoldi framework within physics-informed random projection neural networks that enables reliable computation of the leading eigenvalues governing linear stability and bifurcations of nonlinear PDEs despite the inherent rank deficiency of the collocation discretization.

Recommended citation: Fabiani G., Kavousanakis M.E., Siettos C., Kevrekidis I.G. (2026). Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approach. arXiv preprint arXiv:2603.21568. https://arxiv.org/abs/2603.21568

HEATNETs: Explainable Random Feature Neural Networks for High-Dimensional Parabolic PDEs

Online in Arxiv, 2025

HEATNET, a random feature neural network built from heat-kernel Green’s functions, provides an unbiased and scalable universal approximator for high-dimensional parabolic PDEs, achieving high accuracy up to 2,000 dimensions with relatively few features.

Recommended citation: Georgiou K., Fabiani G., Siettos C., Yannacopoulos A.N. (2025). HEATNETs: Explainable Random Feature Neural Networks for High-Dimensional Parabolic PDEs. arXiv preprint arXiv:2511.00886. https://arxiv.org/abs/2511.00886

Equation-Free Coarse Control of Distributed Parameter Systems via Local Neural Operators

Online in Arxiv, 2025

We propose a framework, using data-driven local neural operators within Krylov subspace methods, to enable efficient model reduction and feedback control of high-dimensional systems without explicit coarse equations.

Recommended citation: Fabiani, G., Siettos, C., & Kevrekidis, I. G. (2025). Equation-Free Coarse Control of Distributed Parameter Systems via Local Neural Operators. arXiv preprint arXiv:2509.23975. https://www.arxiv.org/abs/2509.23975

Going with the Flow: Solving for Symmetry-Driven PDE dynamics with Physics-informed Neural Networks

Online in Arxiv, 2025

We developed a PINN-based framework that factors out continuous symmetries in nonlinear PDEs, transforming them into high-index DAE systems to simultaneously compute invariant solutions and symmetry parameters like wave speeds or scaling rates.

Recommended citation: Kavousanakis, M., Fabiani, G., Georgiou, A., Siettos, C., Kevrekidis, P., & Kevrekidis, I. (2025). Going with the Flow: Solving for Symmetry-Driven PDE dynamics with Physics-informed Neural Networks. arXiv preprint arXiv:2509.15963. https://arxiv.org/abs/2509.15963

Enabling Local Neural Operators to perform Equation-Free System-Level Analysis

Online in Arxiv, 2025

We propose a framework that integrates Neural Operators (NOs) with Krylov subspace iterative methods for efficient stability and bifurcation analysis of large-scale dynamical systems, extending NO applications beyond temporal predictions to system-level numerical tasks

Recommended citation: Fabiani, G., Vandecasteele, H., Goswami, S., Siettos, C., & Kevrekidis, I. G. (2025). Enabling Local Neural Operators to perform Equation-Free System-Level Analysis. arXiv preprint arXiv:2505.02308. https://arxiv.org/abs/2505.02308

Stability Analysis of Physics-Informed Neural Networks for Stiff Linear Differential Equations

Online in Arxiv, 2024

This preprint is about the linear stability analysis and consistency of Phyisics-informed neural networks based on random projections (PIRPNNs)

Recommended citation: Fabiani, G., Bollt, E., Siettos, C., & Yannacopoulos, A. N. (2024). Stability Analysis of Physics-Informed Neural Networks for Stiff Linear Differential Equations. arXiv preprint arXiv:2408.15393. https://doi.org/10.48550/arXiv.2408.15393