Introduction to

Course Syllabus
DISTRIBUTION OF MATERIALS
The slides and source codes are open to individuals with Columbia University email address. If you are interested at learning the materials, please submit a request via the Google drive, preferable with a brief statement explaining the reasons for your interest to the materials (e.g. for educational purpose). The specific term of usage of the materials in this website can be found [here].
CLASS SESSIONS
Guest Lecture: Tuesday and Thursday, 11:30am to 1:20pm at Shriram 104
TEXTBOOK
Supplement materials will be available online.
COURSE DESCRIPTION
This course focuses on a geometric learning approach to derive, test, and validate a wide range of artificial intelligence enabled models for engineering (metamaterials, composites, alloys) and natural materials (soil, rock, clay). Students will learn how to incorporate a wide range of data stored in graphs, manifold and point sets to train neural networks to design optimal experiments, embed highdimensional data, enforce mechanics and physical principles, denoise data with geometry, and enable modelfree simulations and discover causality of mechanisms that leads to the failures of materials.
PREREQUISITES
Linear algebra and mechanics of materials.
ASSESSMENT AND GRADING POLICY
Grades will be based on:
Lab Report 1 ............................................................ 10%
Lab Report 2 ............................................................ 10%
Lab Report 3 ............................................................10%
Lab Report 4 ............................................................ 10%
Quiz 1 .........................................................................10%
Quiz 2 .........................................................................10%
Comprehensive Final Project.................................... 40%
COURSE LOGISTICS
The course will be split into two parts. We will first cover the practical aspect of using neural network to generate models. We will assume that all the data are in the Euclidean space and apply constraints to generate the inductive bias necessary to complete the models. In the second part of the course, we will explore and experiments with different methods to generate constitutive models that are of sufficient robustness and accuracy for engineering practice. For each session, there will be a 1to1.5hours of lecture followed by a COLAB tutorial.
The slides and source codes are open to individuals with Columbia University email address. If you are interested at learning the materials, please submit a request via the Google drive, preferable with a brief statement explaining the reasons for your interest to the materials (e.g. for educational purpose). The specific term of usage of the materials in this website can be found [here].
CLASS SESSIONS
Guest Lecture: Tuesday and Thursday, 11:30am to 1:20pm at Shriram 104
TEXTBOOK
Supplement materials will be available online.
COURSE DESCRIPTION
This course focuses on a geometric learning approach to derive, test, and validate a wide range of artificial intelligence enabled models for engineering (metamaterials, composites, alloys) and natural materials (soil, rock, clay). Students will learn how to incorporate a wide range of data stored in graphs, manifold and point sets to train neural networks to design optimal experiments, embed highdimensional data, enforce mechanics and physical principles, denoise data with geometry, and enable modelfree simulations and discover causality of mechanisms that leads to the failures of materials.
PREREQUISITES
Linear algebra and mechanics of materials.
ASSESSMENT AND GRADING POLICY
Grades will be based on:
Lab Report 1 ............................................................ 10%
Lab Report 2 ............................................................ 10%
Lab Report 3 ............................................................10%
Lab Report 4 ............................................................ 10%
Quiz 1 .........................................................................10%
Quiz 2 .........................................................................10%
Comprehensive Final Project.................................... 40%
COURSE LOGISTICS
The course will be split into two parts. We will first cover the practical aspect of using neural network to generate models. We will assume that all the data are in the Euclidean space and apply constraints to generate the inductive bias necessary to complete the models. In the second part of the course, we will explore and experiments with different methods to generate constitutive models that are of sufficient robustness and accuracy for engineering practice. For each session, there will be a 1to1.5hours of lecture followed by a COLAB tutorial.
Online Course Materials
Date 
Topic 
Lecture 
External Reading 
Assignment 
Tutorial 
1/10 
Course overview, logistic and miscellaneous items 
[Slide] 
N/A 
N/A 

1/12 
Computer Lab: Supervised learning with labeled data: nonlinear regression via neural nets, nonconvex optimization, hyperparameter tuning 
[Colab] 
Google Deepmind X ICL course: [URL] 
Lab Task 1 given 
Value 
1/17 
Supervised learning for elastic energy functional 
[Slide] 
N/A 
MMLDT tutorial: [Video] 

1/19 
Building the learning algorithm with group data: regularization, physics constraints, symmetry, Lie group/Algebra 
[Video] 
N/A 


1/24 
Recurrent neural network for plasticity: Numerical experiments with Long shortterm memory, gated recurrent neural network, and attentionbased transformer. 
[Slide] 
Lab Report 1 due; Lab Task 2 given 
Colab for tractionseparation law: [Colab] 

1/26 
Neural network yield surface and level sets: Theory 
[Slide] 
N/A 

1/31 
Neural network yield surface and level sets: Implementation 
N/A 

2/2 
Physicsinformed neural network as a partial differential equation solver 
[Slide] 
Lab Report 2 due Lab Task 3 given 

2/7 
Overview on nonEuclidean machine learning: metrics, denoising and optimal transport 
Value 
Value 
Value 

2/9 
Geometric Learning Part I: Graph learning in mechanics before machine learning 
N/A 
Embedding of finite element solution [Colab] 

2/14 
Geometric Learning Part I: Graph embedding for nonlinear elasticity (node embedding) 
[Slide] 
Semisupervised learning for a surrogate family [URL] 
Lab Report 3 due 

2/16 
Geometric Learning Part I: Graph embedding and autoencoders for plasticity (graph pooling) 
Graph embedding for plastic deformation: [URL] 
N/A 

2/21 
Geometric Learning Part I: Directed graph, tree, decision tree, and Monte Carlo Tree search for longterm planning 
Value 
Lab Task 4 given 

2/23 
Guest Lecture 1: deep reinforcement learning for the design of experiments Guest Lecture 2: Causal discovery for mechanics 
Value 
N/A 
Value 

2/28 
Geometric Learning Part II: Manifold 
[Slide] 
Lab Report 4 due 

3/2 
Geometric Learning Part II: Modelfree elasticity 
Value 
Value 

3/7 
Geometric Learning Part II: Manifold embedding for denoising 
Value 
N/A 

3/9 
Final project presentations 
N/A 
N/A 


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