Aims

The aims of the course are to:

• Introduce the state-of-the-art concepts and theories for deep learning and neural networks.
• Describe the main methods of constructing learning-based partial differential equation solvers with illustrative examples on Darcy flow and elasticity.
• Explain the concept and theory of path dependency (memory) and multi-scale modelling, with application of the data-driven methods for discovering and approximating constitutive models for various materials.

Objectives

As specific objectives, by the end of the course students should be able to:

• Understand the principles of applying data-driven methods to physical problems.
• Design, implement and train learning-based PDE solvers for stress analysis.
• Discover non-linear, path-dependent material models from data using deep neural networks.

Content

Mechanics and materials are gradually becoming data-rich due to rapid advances in experimental science and high-performance multiscale computing. There has been a growing interest in the field of solid mechanics for developing data-driven and learning-based methods to characterize, understand, model, and design material/structural systems. With data-driven approaches, it is possible to remove/relax the need for ad hoc constitutive models for describing the material behavior, to achieve fast multi-scale computation for structures as well as to generate optimal designs. This module will introduce a wide spectrum of data-driven/learning based methods that have been developed and used in mechanics and materials, with an emphasis on developing a working understanding of how to apply these methods in practice.

Syllabus

Neural network basics (4L)

1. Basic concepts in supervised and unsupervised learning.
2. Fully connected neural network, stochastic gradient descent.
3. Advanced neural network architectures: convolution neural network, Res-net, U-net.
4. Python for machine learning and pytorch tutorial.

Machine learning for PDEs: Physics Informed Neural Networks and Neural Operators (4L)

1. Physics informed neural networks for ODE and PDE.
2. Learning the solution operator of PDE with Neural Operators.
3. Fourier and Graph Neural Operators.

Machine learning for path dependent problems and learning based multi scale modeling (4L))

1. Machine learning methods for memory and path dependence.
2. Long Short Term Memory and Transformer networks.
3. Multiscale modeling and Recurrent Neural Operator.
4. Generative modeling methods.

Data-driven methods in mechanics and beyond - guest lecture (1L)

1. Neural operators in climate change - the earth 2 project.
2. Researches in NVDIA.