Gianluca Fabiani | Università degli Studi di Napoli "Federico II" (original) (raw)

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Papers by Gianluca Fabiani

Research paper thumbnail of Tipping points in overturning circulation mediated by ocean mixing and the configuration and magnitude of the hydrological cycle: A simple model

Journal of physical oceanography, Apr 19, 2024

Research paper thumbnail of Task-oriented machine learning surrogates for tipping points of agent-based models

Nature communications, May 15, 2024

Research paper thumbnail of Review of: "Taylor Series Based Domain Collocation Meshless Method for Problems with Multiple Boundary Conditions including Point Boundary Conditions

Research paper thumbnail of Tipping points in overturning circulation mediated by ocean mixing and the configuration and magnitude of the hydrological cycle: A simple model

arXiv (Cornell University), Aug 7, 2023

Research paper thumbnail of Discrete-time nonlinear feedback linearization via physics-informed machine learning

Journal of Computational Physics

Research paper thumbnail of Discrete-Time Nonlinear Feedback Linearization via Physics-Informed Machine Learning

arXiv (Cornell University), Mar 15, 2023

Research paper thumbnail of Numerical Bifurcation Analysis of PDEs From Lattice Boltzmann Model Simulations: a Parsimonious Machine Learning Approach

Journal of Scientific Computing

We address a three-tier data-driven approach for the numerical solution of the inverse problem in... more We address a three-tier data-driven approach for the numerical solution of the inverse problem in Partial Differential Equations (PDEs) and for their numerical bifurcation analysis from spatio-temporal data produced by Lattice Boltzmann model simulations using machine learning. In the first step, we exploit manifold learning and in particular parsimonious Diffusion Maps using leave-one-out cross-validation (LOOCV) to both identify the intrinsic dimension of the manifold where the emergent dynamics evolve and for feature selection over the parameter space. In the second step, based on the selected features, we learn the right-hand-side of the effective PDEs using two machine learning schemes, namely shallow Feedforward Neural Networks (FNNs) with two hidden layers and single-layer Random Projection Networks (RPNNs), which basis functions are constructed using an appropriate random sampling approach. Finally, based on the learned black-box PDE model, we construct the corresponding bif...

Research paper thumbnail of Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEs

We address a physics-informed neural network based on the concept of random projections for the n... more We address a physics-informed neural network based on the concept of random projections for the numerical solution of IVPs of nonlinear ODEs in linear-implicit form and index-1 DAEs, which may also arise from the spatial discretization of PDEs. The scheme has a single hidden layer with appropriately randomly parametrized Gaussian kernels and a linear output layer, while the internal weights are fixed to ones. The unknown weights between the hidden and output layer are computed by Newton's iterations, using the Moore-Penrose pseudoinverse for low to medium, and sparse QR decomposition with regularization for medium to large scale systems. To deal with stiffness and sharp gradients, we propose a variable step size scheme for adjusting the interval of integration and address a continuation method for providing good initial guesses for the Newton iterations. Based on previous works on random projections, we prove the approximation capability of the scheme for ODEs in the canonical form and index-1 DAEs in the semiexplicit form. The optimal bounds of the uniform distribution are parsimoniously chosen based on the bias-variance trade-off. The performance of the scheme is assessed through seven benchmark problems: four index-1 DAEs, the Robertson model, a model of five DAEs describing the motion of a bead, a model of six DAEs describing a power discharge control problem, the chemical Akzo Nobel problem and three stiff problems, the Belousov-Zhabotinsky, the Allen-Cahn PDE and the Kuramoto-Sivashinsky PDE. The efficiency of the scheme is compared with three solvers ode23t, ode23s, ode15s of the MATLAB ODE suite. Our results show that the proposed scheme outperforms the stiff solvers in several cases, especially in regimes where high stiffness or sharp gradients arise in terms of numerical accuracy, while the computational costs are for any practical purposes comparable.

Research paper thumbnail of Constructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approach

We address a three-tier computational approach for the construction of coarse-grained bifurcation... more We address a three-tier computational approach for the construction of coarse-grained bifurcation diagrams from spatio-temporal data produced by microscopic simulators using machine learning. In the first step, we exploit manifold learning and in particular parsimonious Diffusion Maps to identify the intrinsic dimension of the manifolds where the emergent dynamics evolve and feature selection for the parametrization of these manifolds. In the second step, based on the selected features we learn the right-hand-side of the effective partial differential equations (PDEs) using two machine learning schemes, namely Feed-forward Neural Networks (FNNs) and Random Projection Networks (RPNNs). Finally, based on the learned black-box PDE model, we construct the corresponding bifurcation diagram, thus exploiting numerical bifurcation theory algorithms. For our illustrations, we implemented the proposed method to construct the one-parameter bifurcation diagram of the 1D FitzHugh-Nagumo PDEs fro...

Research paper thumbnail of Numerical Solution of Stiff Odes with Physics Informed RPNNS

arXiv: Numerical Analysis, Nov 24, 2021

Research paper thumbnail of Extreme learning machine collocation for the numerical solution of elliptic PDEs with sharp gradients

We introduce a new numerical method based on machine learning to approximate the solution of elli... more We introduce a new numerical method based on machine learning to approximate the solution of elliptic partial differential equations with collocation using a set of sigmoidal functions. We show that a feedforward neural network with a single hidden layer with sigmoidal functions and fixed, random, internal weights and biases can be used to compute accurately a collocation solution. The choice to fix internal weights and bias leads to the so-called Extreme Learning Machine network. We discuss how to determine the range for both internal weights and biases in order to obtain a good underlining approximating space, and we explore the required number of collocation points. We demonstrate the efficiency of the proposed method with several one-dimensional diffusion-advection-reaction problems that exhibit steep behaviors, such as boundary layers. The boundary conditions are imposed directly as collocation equations. We point out that there is no need of training the network, as the propos...

Research paper thumbnail of Numerical solution and bifurcation analysis of nonlinear partial differential equations with extreme learning machines

Journal of Scientific Computing

We address a new numerical method based on a class of machine learning methods, the so-called Ext... more We address a new numerical method based on a class of machine learning methods, the so-called Extreme Learning Machines (ELM) with both sigmoidal and radial-basis functions, for the computation of steady-state solutions and the construction of (one-dimensional) bifurcation diagrams of nonlinear partial differential equations (PDEs). For our illustrations, we considered two benchmark problems, namely (a) the one-dimensional viscous Burgers with both homogeneous (Dirichlet) and non-homogeneous boundary conditions, and, (b) the one- and two-dimensional Liouville–Bratu–Gelfand PDEs with homogeneous Dirichlet boundary conditions. For the one-dimensional Burgers and Bratu PDEs, exact analytical solutions are available and used for comparison purposes against the numerical derived solutions. Furthermore, the numerical efficiency (in terms of numerical accuracy, size of the grid and execution times) of the proposed numerical machine-learning method is compared against central finite differe...

Research paper thumbnail of Extreme learning machine collocation for the numerical solution of elliptic PDEs with sharp gradients

Computer Methods in Applied Mechanics and Engineering

Research paper thumbnail of Tipping points in overturning circulation mediated by ocean mixing and the configuration and magnitude of the hydrological cycle: A simple model

Journal of physical oceanography, Apr 19, 2024

Research paper thumbnail of Task-oriented machine learning surrogates for tipping points of agent-based models

Nature communications, May 15, 2024

Research paper thumbnail of Review of: "Taylor Series Based Domain Collocation Meshless Method for Problems with Multiple Boundary Conditions including Point Boundary Conditions

Research paper thumbnail of Tipping points in overturning circulation mediated by ocean mixing and the configuration and magnitude of the hydrological cycle: A simple model

arXiv (Cornell University), Aug 7, 2023

Research paper thumbnail of Discrete-time nonlinear feedback linearization via physics-informed machine learning

Journal of Computational Physics

Research paper thumbnail of Discrete-Time Nonlinear Feedback Linearization via Physics-Informed Machine Learning

arXiv (Cornell University), Mar 15, 2023

Research paper thumbnail of Numerical Bifurcation Analysis of PDEs From Lattice Boltzmann Model Simulations: a Parsimonious Machine Learning Approach

Journal of Scientific Computing

We address a three-tier data-driven approach for the numerical solution of the inverse problem in... more We address a three-tier data-driven approach for the numerical solution of the inverse problem in Partial Differential Equations (PDEs) and for their numerical bifurcation analysis from spatio-temporal data produced by Lattice Boltzmann model simulations using machine learning. In the first step, we exploit manifold learning and in particular parsimonious Diffusion Maps using leave-one-out cross-validation (LOOCV) to both identify the intrinsic dimension of the manifold where the emergent dynamics evolve and for feature selection over the parameter space. In the second step, based on the selected features, we learn the right-hand-side of the effective PDEs using two machine learning schemes, namely shallow Feedforward Neural Networks (FNNs) with two hidden layers and single-layer Random Projection Networks (RPNNs), which basis functions are constructed using an appropriate random sampling approach. Finally, based on the learned black-box PDE model, we construct the corresponding bif...

Research paper thumbnail of Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEs

We address a physics-informed neural network based on the concept of random projections for the n... more We address a physics-informed neural network based on the concept of random projections for the numerical solution of IVPs of nonlinear ODEs in linear-implicit form and index-1 DAEs, which may also arise from the spatial discretization of PDEs. The scheme has a single hidden layer with appropriately randomly parametrized Gaussian kernels and a linear output layer, while the internal weights are fixed to ones. The unknown weights between the hidden and output layer are computed by Newton's iterations, using the Moore-Penrose pseudoinverse for low to medium, and sparse QR decomposition with regularization for medium to large scale systems. To deal with stiffness and sharp gradients, we propose a variable step size scheme for adjusting the interval of integration and address a continuation method for providing good initial guesses for the Newton iterations. Based on previous works on random projections, we prove the approximation capability of the scheme for ODEs in the canonical form and index-1 DAEs in the semiexplicit form. The optimal bounds of the uniform distribution are parsimoniously chosen based on the bias-variance trade-off. The performance of the scheme is assessed through seven benchmark problems: four index-1 DAEs, the Robertson model, a model of five DAEs describing the motion of a bead, a model of six DAEs describing a power discharge control problem, the chemical Akzo Nobel problem and three stiff problems, the Belousov-Zhabotinsky, the Allen-Cahn PDE and the Kuramoto-Sivashinsky PDE. The efficiency of the scheme is compared with three solvers ode23t, ode23s, ode15s of the MATLAB ODE suite. Our results show that the proposed scheme outperforms the stiff solvers in several cases, especially in regimes where high stiffness or sharp gradients arise in terms of numerical accuracy, while the computational costs are for any practical purposes comparable.

Research paper thumbnail of Constructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approach

We address a three-tier computational approach for the construction of coarse-grained bifurcation... more We address a three-tier computational approach for the construction of coarse-grained bifurcation diagrams from spatio-temporal data produced by microscopic simulators using machine learning. In the first step, we exploit manifold learning and in particular parsimonious Diffusion Maps to identify the intrinsic dimension of the manifolds where the emergent dynamics evolve and feature selection for the parametrization of these manifolds. In the second step, based on the selected features we learn the right-hand-side of the effective partial differential equations (PDEs) using two machine learning schemes, namely Feed-forward Neural Networks (FNNs) and Random Projection Networks (RPNNs). Finally, based on the learned black-box PDE model, we construct the corresponding bifurcation diagram, thus exploiting numerical bifurcation theory algorithms. For our illustrations, we implemented the proposed method to construct the one-parameter bifurcation diagram of the 1D FitzHugh-Nagumo PDEs fro...

Research paper thumbnail of Numerical Solution of Stiff Odes with Physics Informed RPNNS

arXiv: Numerical Analysis, Nov 24, 2021

Research paper thumbnail of Extreme learning machine collocation for the numerical solution of elliptic PDEs with sharp gradients

We introduce a new numerical method based on machine learning to approximate the solution of elli... more We introduce a new numerical method based on machine learning to approximate the solution of elliptic partial differential equations with collocation using a set of sigmoidal functions. We show that a feedforward neural network with a single hidden layer with sigmoidal functions and fixed, random, internal weights and biases can be used to compute accurately a collocation solution. The choice to fix internal weights and bias leads to the so-called Extreme Learning Machine network. We discuss how to determine the range for both internal weights and biases in order to obtain a good underlining approximating space, and we explore the required number of collocation points. We demonstrate the efficiency of the proposed method with several one-dimensional diffusion-advection-reaction problems that exhibit steep behaviors, such as boundary layers. The boundary conditions are imposed directly as collocation equations. We point out that there is no need of training the network, as the propos...

Research paper thumbnail of Numerical solution and bifurcation analysis of nonlinear partial differential equations with extreme learning machines

Journal of Scientific Computing

We address a new numerical method based on a class of machine learning methods, the so-called Ext... more We address a new numerical method based on a class of machine learning methods, the so-called Extreme Learning Machines (ELM) with both sigmoidal and radial-basis functions, for the computation of steady-state solutions and the construction of (one-dimensional) bifurcation diagrams of nonlinear partial differential equations (PDEs). For our illustrations, we considered two benchmark problems, namely (a) the one-dimensional viscous Burgers with both homogeneous (Dirichlet) and non-homogeneous boundary conditions, and, (b) the one- and two-dimensional Liouville–Bratu–Gelfand PDEs with homogeneous Dirichlet boundary conditions. For the one-dimensional Burgers and Bratu PDEs, exact analytical solutions are available and used for comparison purposes against the numerical derived solutions. Furthermore, the numerical efficiency (in terms of numerical accuracy, size of the grid and execution times) of the proposed numerical machine-learning method is compared against central finite differe...

Research paper thumbnail of Extreme learning machine collocation for the numerical solution of elliptic PDEs with sharp gradients

Computer Methods in Applied Mechanics and Engineering