Federico Pichi

International School for Advanced Studies | SISSA · Applied Mathematics Group

This work deals with the investigation of bifurcating fluid phenomena using a reduced order modelling setting aided by artificial neural networks. We discuss the POD-NN approach dealing with non-smooth solutions set of nonlinear parametrized PDEs. Thus, we study the Navier-Stokes equations describing: (i) the Coanda effect in a channel, and (ii) th...

This work deals with optimal control problems as a strategy to drive bifurcating solution of nonlinear parametrized partial differential equations towards a desired branch. Indeed, for these governing equations, multiple solution configurations can arise from the same parametric instance. We thus aim at describing how optimal control allows to chan...

The present work proposes a framework for nonlinear model order reduction based on a Graph Convolutional Autoencoder (GCA-ROM). In the reduced order modeling (ROM) context, one is interested in obtaining real-time and many-query evaluations of parametric Partial Differential Equations (PDEs). Linear techniques such as Proper Orthogonal Decompositio...

Reduced order models (ROMs) are widely used in scientific computing to tackle high-dimensional systems. However, traditional ROM methods may only partially capture the intrinsic geometric characteristics of the data. These characteristics encompass the underlying structure, relationships, and essential features crucial for accurate modeling. To ov...

This work presents a novel resolution-invariant model order reduction strategy for multifidelity applications. We base our architecture on a novel neural network layer developed in this work, the graph feedforward network, which extends the concept of feedforward networks to graph-structured data by creating a direct link between the weights of a n...

We present a novel reduced-order Model (ROM) that leverages optimal transport (OT) theory and displacement interpolation to enhance the representation of nonlinear dynamics in complex systems. While traditional ROM techniques face challenges in this scenario, especially when data (i.e., observational snapshots) is limited, our method addresses thes...

This paper presents a projection-based reduced order modelling (ROM) framework for unsteady parametrized optimal control problems (OCP$_{(\mu)}$s) arising from cardiovascular (CV) applications. In real-life scenarios, accurately defining outflow boundary conditions in patient-specific models poses significant challenges due to complex vascular morp...

Incorporating probabilistic terms in mathematical models is crucial for capturing and quantifying uncertainties of real-world systems. However, stochastic models typically require large computational resources to produce meaningful statistics. For such reason, the development of reduction techniques becomes essential for enabling efficient and scal...

In this paper, we discuss reduced order modelling approaches to bifurcating systems arising from continuum mechanics benchmarks. The investigation of the beam’s deflection is a relevant topic of investigation with fundamental implications on their design for structural analysis and health. When the beams are exposed to external forces, their equili...

In this paper, we introduce the neural empirical interpolation method (NEIM), a neural network-based alternative to the discrete empirical interpolation method for reducing the time complexity of computing the nonlinear term in a reduced order model (ROM) for a parameterized nonlinear partial differential equation. NEIM is a greedy algorithm which...

In this chapter, we consider a scalar advection-diffusion PDE modeling the Graetz flow problem in a two-dimensional and geometrically parametrized domain. The aim is to obtain efficient evaluations of the thermal field and the output of interest, defined as the integral of the temperature over the outflow boundary. The solution is approximated usin...

In this chapter, we consider a vector elliptic PDE modeling a parametric linear elasticity problem in a two-dimensional heterogeneous domain. The aim is to obtain efficient evaluations of the vector displacement field and the output of interest, defined as the integrated horizontal displacement over the loaded boundary. The solution is approximated...

In this chapter, we introduce the readers to the main notions regarding the Reduced Basis approximation based on Finite Element method for parametrized Partial Differential Equations. We recall definitions and techniques useful to understand and analyze the application of Reduced Order Modeling to the worked out problems discussed in the chapters o...

In this chapter, we consider a scalar elliptic PDE modeling a parametric steady heat conduction problem in a two-dimensional domain. The aim is to obtain efficient evaluations of the thermal field and the output of interest, defined as the average temperature over the boundary. The solution is approximated using reduced order modeling techniques ba...

In this chapter, we consider a scalar elliptic PDE modeling a steady-state heat conduction problem in a two-dimensional and geometrically parametrized holed domain. The aim is to obtain efficient evaluations of the thermal field and the output of interest, defined as the average conduction temperature distribution at the inner walls. The solution i...

In this chapter, we consider a vector elliptic PDE modeling a linear elasticity problem for bridge designing in a two-dimensional and geometrically parametrized domain. The aim is to obtain efficient evaluations of the vector displacement field and the output of interest, defined as the integrated vertical displacement over the loaded boundary. The...

In this chapter, we consider a scalar advection-diffusion PDE modeling a stabilized advection dominated flow problem in a two-dimensional domain. The aim is to obtain efficient evaluations of the unknown field for high Péclet number and the output of interest, defined as the average temperature over the boundary. The solution is approximated using...

In this chapter, we consider a parabolic PDE modeling the unsteady conduction problem in a two-dimensional domain. The aim is to obtain efficient evaluations of the time evolution of the thermal field and the output of interest, defined as the average temperature over the boundary. The solution is approximated using reduced order modeling technique...

In this chapter, we consider a vector steady nonlinear PDE modeling the bifurcating Coanda effect for an incompressible flow in a two-dimensional channel. The aim is to obtain efficient evaluations of the velocity and pressure fields and the output of interest, defined as the integral of vertical velocity, which is related to the symmetry breaking...

In this chapter, we consider a scalar elliptic PDE modeling the steady heat conduction problem with a non-affine Gaussian flux in a two-dimensional domain. The aim is to obtain efficient evaluations of the thermal field and the output of interest, defined as the integral of the source term. The solution is approximated using reduced order modeling...

In this chapter, we consider a vector time-dependent nonlinear PDE modeling the unsteady Navier-Stokes system for an incompressible flow in a two-dimensional domain with a cylindrical obstacle. The aim is to obtain efficient evaluations of the velocity and pressure fields and the output of interest, defined as the average of the solution over the w...

In this chapter, we consider a linear-quadratic optimal control problem with a vector elliptic PDE constraint in a two-dimensional and geometrically parametrized domain. The aim is to obtain efficient evaluations of the optimal temperature distribution, the adjoint, the optimal control field and the output of interest, defined as the cost functiona...

In this chapter, we consider a vector nonlinear parabolic PDE modeling the unsteady FitzHugh–Nagumo system in a one-dimensional domain. The aim is to obtain efficient evaluations of the unknown fields describing the voltage and the recovery voltage. The solution is approximated using reduced order modeling techniques based on the POD-Galerkin metho...

In this chapter, we consider a vector elliptic PDE modeling a contact problem in linear elasticity with friction in a two-dimensional and geometrically parametrized domain. The aim is to obtain efficient evaluations of the vector displacement field and the output of interest, defined as the integrated vertical displacement over the loaded boundary....

In this chapter, we consider a vector elliptic stochastic PDE modeling a steady heat conduction in a two-dimensional heterogeneous domain based on random input data. The aim is to obtain efficient evaluation of the statistics for the thermal field and the output of interest, defined as the average temperature over the whole domain. The solution is...

In this chapter, we consider a scalar nonlinear elliptic PDE as a benchmark for complex material problems in a two-dimensional domain. The aim is to obtain efficient evaluations of the unknown field and the output of interest, defined as its average over the whole domain. The solution is approximated using reduced order modeling techniques based on...

In this chapter, we consider a vector steady nonlinear PDE modeling the Navier–Stokes system for an incompressible flow in a two-dimensional domain. The aim is to obtain efficient evaluations of the velocity and pressure fields and the output of interest, defined as the average of the solution over the whole domain. The solution is approximated usi...

In this chapter, we consider a scalar elliptic PDE modeling the steady-state heat transfer problem through a two-dimensional geometrically parametrized fin. The aim is to obtain efficient evaluations of the thermal field and the output of interest, defined as a measure of its dissipative capabilities. The solution is approximated using reduced orde...

Reduced order models (ROMs) are widely used in scientific computing to tackle high-dimensional systems. However, traditional ROM methods may only partially capture the intrinsic geometric characteristics of the data. These characteristics encompass the underlying structure, relationships, and essential features crucial for accurate modeling. To ov...

In this paper, we discuss reduced order modelling approaches to bifurcating systems arising from continuum mechanics benchmarks. The investigation of the beam's deflection is a relevant topic of investigation with fundamental implications on their design for structural analysis and health. When the beams are exposed to external forces, their equili...

This work explores the development and the analysis of an efficient reduced order model for the study of a bifurcating phenomenon, known as the Coandă effect, in a multi‐physics setting involving fluid and solid media. The latter is governed by the Navier‐Stokes equations for an incompressible, steady and viscous fluid and by the elasticity constit...

This work explores the development and the analysis of an efficient reduced order model for the study of a bifurcating phenomenon, known as the Coand\u{a} effect, in a multi-physics setting involving fluid and solid media. Taking into consideration a Fluid-Structure Interaction problem, we aim at generalizing previous works towards a more reliable...

The aim of this work [1] is to show the applicability of the Reduced Basis (RB) model reduction and Artificial Neural Network (ANN) dealing with parametrized Partial Differential Equations (PDEs) in nonlinear systems undergoing bifurcations. Bifurcation analysis, i.e., following the different bifurcating branches due to the non‐uniqueness of the so...

The majority of the most common physical phenomena can be described using partial differential equations (PDEs). However, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical metho...

This work deals with optimal control problems as a strategy to drive bifurcating solutions of nonlinear parametrized partial differential equations towards a desired branch. Indeed, for these governing equations, multiple solution configurations can arise from the same parametric instance. We thus aim at describing how optimal control allows to cha...

We propose a computationally efficient framework to treat nonlinear partial differential equations having bifurcating solutions as one or more physical control parameters are varied. Our focus is on steady bifurcations. Plotting a bifurcation diagram entails computing multiple solutions of a parametrized, nonlinear problem, which can be extremely e...

The majority of the most common physical phenomena can be described using partial differential equations (PDEs), however, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical metho...

This work focuses on the detection of the buckling phenomena and bifurcation analysis of the parametric Von K\'arm\'an plate equations based on reduced order methods and spectral analysis. The computational complexity - due to the fourth order derivative terms, the non-linearity and the parameter dependence - provides an interesting benchmark to te...

The aim of this work is to show the applicability of the Reduced Basis (RB) model reduction in nonlinear systems undergoing bifurcations. Bifurcation analysis, i.e. following the different bifurcating branches and determining the bifurcation points, is a complex computational task. Reduced Order Models (ROM) can reduce the computational burden, ena...

In this work we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for elasticity problems in affinely parametrized geometries. The essential ingredients of the methodology are: a Galerkin projection onto a low-dimensional space associated with a smooth “parametric manifold”—dimension reduction; an effic...

The aim of this work is to show the applicability of the reduced basis model reduction in nonlinear systems undergoing bifurcations. Bifurcation analysis, i.e., following the different bifurcating branches, as well as determining the bifurcation point itself, is a complex computational task. Reduced Order Models (ROM) can potentially reduce the com...

This work focuses on the detection of the buckling phenomena and bifurcation analysis of the parametric Von K\'arm\'an plate equations based on reduced order methods and spectral analysis. The computational complexity - due to the fourth order derivative terms, the non-linearity and the parameter dependence - provides an interesting benchmark to te...

In this work we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for elasticity problems in affinley parametrized geometries. The essential ingredients of the methodology are: a Galerkin projection onto a low-dimensional space associated with a smooth "parametric manifold" - dimension reduction, an eff...