FAU MoD workshop L. Liverani / H. Holthusen

Date: Tue. September 23, 2025
Event: FAU MoD Lecture
Organized by: FAU MoD, the Research Center for Mathematics of Data at Friedrich-Alexander-Universität Erlangen-Nürnberg (Germany)

Session 01: 11:00H
HYCO: Hybrid-Cooperative Learning for PDEs
Speaker: Dr. Lorenzo Liverani
Affiliation: FAU MoD, Research Center for Mathematics of Data | FAU DCN-AvH at Friedrich-Alexander-Universität Erlangen-Nürnberg
Abstract. In this talk, I will present the Hybrid-Cooperative Learning strategy (HYCO), a modeling framework that iteratively integrates physics-based and data-driven models through a mutual regularization mechanism.

Unlike traditional approaches that impose physical constraints directly on synthetic models, HYCO treats the physical and synthetic components as co-trained agents, nudging them towards mutual agreement. This cooperative learning scheme is naturally parallelizable and improves robustness to noise as well as to sparse or heterogeneous data. Through several numerical experiments on both static and time-dependent problems, I will demonstrate that HYCO is a promising architecture, outperforming classical physics-based and data-driven methods and recovering accurate solutions and model parameters even under ill-posed conditions.
This is ongoing work with E. Zuazua and M. Steynberg.

Session 02: 11:30H
A Neural Network Companion to Inelasticity
Speaker: Dr.-Ing. Hagen Holthusen
Affiliation: FAU MoD, Research Center for Mathematics of Data | Institute of Applied Mechanics at Friedrich-Alexander-Universität Erlangen-Nürnberg (Germany)

Abstract. Neural networks have become ubiquitous in science, helping us uncover patterns in complex data sets — from social interactions to medical image analysis.
Yet, their reliability often hinges on large amounts of training data, which is typically scarce in mechanics due to the high cost of experiments. Moreover, we are frequently interested in predicting the mechanical behaviour of solids beyond the range of observed data, a regime where conventional neural networks tend to perform poorly. To address these challenges, the community has proposed various strategies, with physics-informed neural networks being among the most widely discussed.

In this talk, however, we focus on physics-embedded neural networks for discovering the mechanics of solids that exhibit dissipative, irreversible behaviour — that is, where part of the stored energy cannot be recovered. Unlike physics-informed networks, which incorporate physical laws into the loss function, physics-embedded networks are architecturally constrained so that uncertainty-free physical principles are satisfied by design. This architectural integration can improve generalisation, even when data are limited.

Importantly, even within the same material class, inelastic behaviour can differ significantly — aluminium and titanium, for instance, show markedly different irreversible responses. This highlights the crucial trade-off between the interpretability and expressivity of neural networks when modelling complex, inelastic materials: a network must be flexible enough to capture material-specific features, yet structured enough to remain interpretable and physically meaningful.

OUR SPEAKERS

Lorenzo Liverani has completed his PhD in 2023 under the supervision of prof. Vittorino Pata in Politecnico di Milano. He has a background in the long time behavior of partial differential equations and semigroup theory. After a one-year postdoc at Università Milano Bicocca, under the supervision of prof. Veronica Felli, he joined the DCN-AvH chair in February 2024 as a Humboldt Fellow. Now he works at the interface of machine learning and differential equations, focusing on Neural ODE architectures and hybrid techniques for PDE modeling.

AUDIENCE

This is a hybrid event (On-site/online) open to: Public, Students, Postdocs, Professors, Faculty, Alumni and the scientific community all around the world.