METaL Seminars

Ambrogio Maria Bernardelli

Optimazing AI: MILPs for discrete Neural Networks

Training neural networks is typically done using gradient-based methods, which often require large datasets, significant computational resources, and careful hyperparameter tuning. In this presentation, we explore an alternative approach based on Mixed-Integer Linear Programming (MILP) to train discrete neural networks exactly, particularly in low-data settings. The focus is on few-bit neural networks, including Binarized Neural Networks (BNNs), whose weights are restricted to +1 and −1, and Integer-Valued Neural Networks (INNs), whose weights lie within a limited integer range {−P, ..., P}. These models are especially attractive because of their lightweight architecture and their ability to run on low-power devices, where computations can be implemented using simple Boolean or integer operations. A new multiobjective ensemble method, called BeMi, is introduced. Instead of training a single network to distinguish all classes, the approach trains one network for each pair of classes and combines their predictions using a majority voting scheme. The training process simultaneously optimizes accuracy, robustness to small input perturbations, and sparsity, reducing the number of active weights in the network. Experimental results on the MNIST dataset show significant improvements over previous solver-based approaches. While earlier methods achieved an average accuracy of 51.1%, the proposed ensemble method reaches 68.4% accuracy when trained with 10 images per class and 81.8% accuracy with 40 images per class. At the same time, it removes up to 75.3% of the network connections, producing simpler and more efficient models. 

Luigi Greco

High-order Phase-Field Models for accurate and efficient 

fracture simulations via Isogeometric discretization: 

theoretical aspects and experimental applications.

Fracture phenomena can compromise the functionality of structural elements and industrial components, making their prevention a fundamental design requirement. Phase-field models have become increasingly popular for simulating complex crack propagation, thanks to their versatility and robustness. However, standard phase-field approaches often come with high computational costs.

This talk presents high-order phase-field models for fracture, which allow for a reduction of the computational cost by exploiting the increased regularity of the problem. Isogeometric Analysis is employed as a discretization framework, enabling an efficient treatment of high-order terms. The approach is assessed through a numerical-experimental comparison, aimed at reproducing the observed crack patterns and key engineering quantities of interest, thus demonstrating its ability to capture relevant fracture behaviour in practical applications.

Gabriele Loli

Efficient solvers for Isogeometric Analysis

We discuss efficient linear solvers for PDEs discretized through Isogeometric Analysis (IGA). In IGA, spline and Non-Uniform Rational B-Spline (NURBS) bases are used both to describe the geometry and to build the approximation spaces, allowing one to work with high-order smooth functions while keeping an exact representation of the domain. The methods presented here rely on the tensor-product structure of multivariate B-splines. In particular, we exploit the Kronecker-product structure of the matrices arising from Galerkin discretizations to construct preconditioners for several classes of PDEs.

For elliptic and parabolic problems, we use Fast Diagonalization (together with some extensions) to invert preconditioners expressed as sums of Kronecker products of univariate matrices. For hyperbolic problems, in particular the wave equation, we instead turn to Schur decompositions, which allow us to design solvers that avoid assembling very large space–time matrices.

A set of numerical experiments will be shown to illustrate the performance of the proposed strategies.

Antonio Maria D'Altri

Computational Analysis of Masonry and Heritage Structures:

recent advances across aifferent scales

This seminar provides an overview of computational modeling techniques applied to masonry and heritage structures, emphasizing recent advances across different scales, i.e., from a full-scale monument (e.g., medieval castle) down to the micrometer scale of a single pore in porous building material. The presentation begins by addressing the structural scale, focusing on the numerical modeling and structural analysis of full-scale heritage structures. A key challenge discussed is finite element (FE) mesh generation from point clouds, featuring procedures such as Cloud2FEM. This is followed by an overview of analysis approaches for conducting structural assessments on large-scale historical structures. Next, the focus narrows to a high-fidelity block-based mechanical model for masonry, where damaging blocks interact through cohesive-frictional contact constraints. Applications, including those in the dynamic regime under earthquake-like actions, will be shown. A recent advancement is the "continualization" of this block-based model through deriving a mechanism-based strength domain and implementing it within plasticity and damage computational frameworks. Finally, the discussion shifts to the micrometer scale with the simulation of salt crystallization-induced damage in porous materials. This involves multiphase modeling of water and salt transport within porous building materials, coupled with micromechanics-based damage estimations. The seminar will conclude with a brief mention of advances in the Virtual Element Method (VEM) and an overview of the Marie Skłodowska-Curie Postdoctoral Fellowships (MSCA PF) call, highlighting this opportunity for future research funding for young investigators.

Matteo Negri

From Energy Functionals to Cracks: 

bridging analysis and computation in phase-field fracture

We present a general approach for quasi-static and dynamic fracture propagation based on energy functionals. Quasi-static (steady-state) evolutions are governed by energy identity and equilibrium, with respect to displacement and crack variations. We present these basic ingredients and the resulting system of evolution equations for a toy sharp-crack model and then for a phase-field model. In the latter case the evolution is computed by means of constrained incremental problems, which we solve using a staggered algorithm. Consistency between sharp and phase-field approaches is supported by Γ-convergence and numerical simulations, however the overall theoretical and computational picture is far from being complete and raises open problems on convergence of energies, algorithms and evolutions. For dynamic problems, evolutions are governed by energy identity and equilibrium (as in quasi-static) together with stationary action. Again, we compute phase-field evolutions by a staggered scheme where the wave equation is solved by a Newmark method, in a way that numerical dissipation is negligible; as a result simulations feature energy conservation and a precise representation of P- and S-waves. In this setting, high order phase-field energies turn out to be crucial to reduce the computational cost without compromising accuracy.