Background:
For more than 100 years, chemical, physical, and material scientists have proposed competing constitutive models to best characterize the behaviour of man-made and natural materials in response to mechanical loading. Now, computer science offers a universal solution: neural networks. Neural networks are powerful function approximators that learn constitutive relations from big data without any knowledge of the underlying physics. However, classical neural networks entirely ignore a century of research in constitutive modelling, violate thermodynamic considerations, and fail to predict the behaviour outside the training regime. In this pre-course, we introduce constitutive neural networks, a new family of neural networks that inherently satisfy common kinematic, thermodynamic, and physical constraints and, at the same time, constrain the design space of admissible functions to create robust approximators, even in the presence of sparse data. We show that this new class of network models is a generalization of the classical Neo-Hooke, Blatz Ko, Mooney Rivlin, Yeoh, Demiray, and Holzapfel models and that the network weights have a clear physical interpretation in the form of shear moduli, stiffness-like parameters, and exponential coefficients. To familiarize yourself with this new technology, you will implement your own constitutive neural network and train it with classical benchmark data, for example, from brain, skin, arteries, and the heart. You are welcome to bring your own data! You will see that your constitutive neural network autonomously selects the best constitutive model, parameters, and experiment to characterize your material. This new technology could have the potential to induce a paradigm shift in constitutive modelling, from user-defined model selection to automated model discovery. At the end of the course, we will share source codes, benchmark data, and documented examples.

Content and learning objectives:
In this short course, you will learn about automated model discovery; participate in a hands-on programming experience to implement and train your own constitutive neural networks; and receive a library of neural network notebooks to analyse and interpret classical benchmark data of living materials like human brain, skin, arteries, or the heart.

Structure and duration:
1. Brief history of constitutive modelling (20 minutes)
2. Introduction to constitutive neural networks (20 minutes)
3. Overview of mechanical testing (20 minutes)
Short break for discussion and Q&A
4. Automated model discovery for biological systems

    using your own Jupyter notebooks (60 minutes)

Potential attendees:
This pre-course is designed equally for Bachelor’s, Master’s, and PhD students, postdocs, and junior faculty who are interested in mechanical testing, analyzing data, modeling soft biological systems, and using machine learning and artificial intelligence to discover new knowledge from data, or in collaborating with our group and others to push the frontiers in automated model discovery.

Requirements from the participants:
Bring your own laptop if you want to participate in the hands-on automated model discovery part of the course. We will provide Jupyter notebooks, source codes, benchmark data, and documented examples, but you are welcome to bring your own data.

Ellen Kuhl is the Catherine Holman Johnson Director of Stanford Bio-X and the Walter B. Reinhold Professor in the School of Engineering. Her area of expertise is Living Matter Physics. She received the 2023 ERC Advanced Grant for her research on Automated Model Discovery. She is a passionate marathon runner and a multiple Ironman World Championship participant.

Mathias Peirlinck is Assistant Professor of BioMechanical Engineering at Delft University of Technology in the Netherlands. His area of expertise is soft tissue mechanics and in silico cardiovascular modeling. He received the 2023 Veni Talent Award for his research on data-driven and physics-based inference of myocardial tissue behavior.

Skyler St. Pierre is a Mechanical Engineering PhD Candidate in Professor Kuhl’s group. Currently, he conducts experimental tests on plant-based and animal meat and uses physics-based neural networks to model their constitutive behavior. He is funded by the NSF Graduate Research Fellowship and Stanford DARE Fellowship. He is also a triathlete.

Moritz Flaschel holds a PhD Degree in Computational Mechanics, which he acquired in Prof. Laura De Lorenzis’ group at ETH Zürich. In his doctoral thesis, he developed the method EUCLID, a method for automatically discovering interpretable material models from full-field data. Currently, he is a Postdoctoral Researcher in Prof. Ellen Kuhl’s group.