Recent progress in artificial intelligence techniques has delivered tools to solve complex and very high dimensional optimization problems that allow to discover correlations in large data sets, and hence to extract features and patterns from measurements or simulations of complex phenomena.
Applications of these machine learning techniques to the field of solid mechanics range from the identification of constitutive equation from full-field time-dependent measurements, to machine learning techniques for the regularization of ill-posed inverse problems in imaging and control, and to the automatic discovery of reduced order models recapitulating the main features of observed patterns in complex multi-physical systems, thanks to the automatic discovery of hidden structures and symmetries in nonlinear dynamical systems.
Building upon some introductory lectures on the mathematical foundations of machine learning and on the basic computational tools required, these recent and potentially revolutionary approaches to long standing problems in nonlinear solid mechanics will be discussed in the context of specific case studies from engineering applications.
We plan to cover the fundamental mathematical background of machine learning techniques to explain and rationalize the reasons for their success, to illustrate the possibility of automated discovery of dimensionally-reduced hidden structures and symmetries in solid mechanics problems, and to illustrate the potential of machine learning techniques in the context of application to specific classes of material systems ranging from metals, to polymers, to granular materials.
The school will be structured according to the following plan:
Mathematical background to neural networks and neural operators.
Lectures by Carola Schönlieb (Cambridge) and Nikola Kovachki (Nvidia/NYU).
Discovery of internal variables and invariant manifolds in history-dependent phenomena.
Lectures by Antonio DeSimone (SISSA/Pisa) and Kaushik Bhattacharya (Caltech).
Learning from experimental data and connections to numerical methods.
Lectures by Dirk Mohr (ETH) and Laurent Stainier (Nantes).
Gilbert Strang, Linear Algebra and learning form data, Wellesley-Cambridge Press 2019.
Tamara Grossmann et al.: Can physics-informed neural networks beat the finite element method?
IMA Journal Applied Mathematics (2024) 89, 143-174.
Kaushik Bhattacharya et al.: Learning homogenization for elliptic operators, SIAM Journal on Numerical Analysis 62 (4), 1844-1873.
P. Carrara et al.; Data-driven fracture mechanics. Computer Methods in Applied Mechanics and Engineering 372, 113390 (2020).
J. Ulloa et al.: Data-driven micromorphic mechanics for materials with strain localization. arXiv preprint arXiv:2402.15966.
A. Leygue et al.: Data-based derivation of material response. Comput. Methods Appl. Mech. Engrg., 331, pp. 184-196 (2018).
A. Leygue et al.: Non-parametric material state field extraction from full field measurements. Computational Mechanics 64:501-509 (2019).
L. Stainier et al.: Model-free data-driven methods in mechanics: material data identification and solvers. Computational Mechanics 64:381-393 (2019).
5 lectures on:
Learning history-dependent multi-scale phenomena:
History-dependent multi-scale phenomena in continuum mechanics; Internal variables and gradient flows; Recurrent neural operators; Application to experimental learning of constitutive behavior; Application to polycrystalline plasticity.
5 lectures on:
Mesh-free discretization and model reduction in time-dependent problems:
Oscillatory instabilities in the mechanics of solids; Analytical and computational techniques for model reduction in nonlinear dynamical systems; Center manifold theory; Mesh-free methods based on PINNs; Applications to self-oscillating active elastic systems and light-activated Liquid Crystal Elastomers.
6 lectures on:
Learning Solution Operators for the PDEs of Solid Mechanics:
Neural Networks and Deep Learning; Approximation Theory of Neural Networks; Operator Learning; Approximation Theory of Neural Operators; Homogenization of Elliptic Equations; Data-driven Approximation of Crystal Plasticity.
6 lectures on:
Artificial Intelligence techniques in optimal design problems in the Mechanics and Manufacturing of Lightweight Solid Materials.
6 lectures on:
The mathematical bases of machine learning and neural networks and on their use in image analysis and in the solution of inverse problems in the mechanics of solids:
Mathematical operators and applications; Learning an operator from data; Neural networks for operator learning: Fourier neural operator, PINNs; Structure-preserving learning with neural networks; Inverse problems and applications; Learning an inverse operator.
6 lectures on:
The use of machine learning in the extraction of constitutive equations from experimental data:
Constitutive model free data-driven computational mechanics (DDCM): from non-linear elasticity to history-dependent material response; Model free data-driven identification (DDI) from field measurements and elastography; DDCM driven concurrent multiscale simulations and goal-oriented data sampling/discovery. Examples and applications will include nonlinear (hyper-)elasticity, damage, viscoelasticity of polymers, plasticity of granular materials.
ADMISSION AND ACCOMMODATION
The course is offered in a hybrid format, allowing participants the flexibility to attend either in person or remotely via the Microsoft Teams platform.
Admission to on-site attendance is granted on a first-come, first-served basis to comply with the capacity of the lecture room.
Application forms should be submitted online through the website: http://www.cism.it. A confirmation message will be sent to participants whose applications are accepted.
Registration fees:
- Early Bird On-Site Participation: € 650.00 + VAT* - Deadline: July 29, 2025
- Late On-Site Participation: € 800.00 + VAT* - Deadline: September 16, 2025
- Live Streaming Online Participation: € 250.00 + VAT* - Deadline: September 16, 2025
On-site participation includes a complimentary bag, five fixed menu buffet lunches, hot beverages, downloadable lecture notes.
Online participation includes downloadable lecture notes.
Upon request, and subject to availability, a limited number of on-site participants can be accommodated at the CISM Guest House for € 35 per person per night. To request accommodation, please contact foresteria@cism.it.
CANCELLATION POLICY
Applicants may cancel their registration and receive a full refund by notifying the CISM Secretariat in writing (via email) no later than:
- July 29, 2025, for early bird on-site participation;
- August 29, 2025, for late on-site participation;
- September 16, 2025, for online participation.
No refunds after the deadlines. Cancellation requests received before these deadlines and incorrect payments will be subject to a € 50.00 handling fee.
GRANTS
A limited number of participants from universities and research centers who do not receive support from their institutions can request a waiver of the registration fee and/or free lodging.
Requests should be sent to the CISM Secretariat by July 29, 2025, along with the applicant's curriculum vitae and a letter of recommendation from the head of the department or a supervisor confirming that the institute cannot provide funding. Preference will be given to applicants from countries that sponsor CISM.
coming soon courses
CISM-EUROMECH Advanced Course on "Instabilities in Turbulence"
10th CISM-ECCOMAS Advanced course on "Opportunities and Challenges of Quantum Computing in Computational Mechanics"
Wrinkling – Theoretical Foundations, Experimental Characterization and Numerical Modeling
28th CISM-IUTAM Summer School on "Gravity Currents in the Environment"
Crack Initiation and Failure in Complex Structures: Finite Fracture Mechanics and Related Fracture Models