Homework 7 - TEMPLATE - DO NOT DO IT YET

Homework 7 - TEMPLATE - DO NOT DO IT YET#

References#

Module 6: Physics-informed neural networks
- PINNs basics
- PINNs for parametric studies
- PINNs for inverse problems
Module 7: Inverse problems in stochastic scientific models
- Stochastic differential equations

Instructions#

Type your name and email in the “Student details” section below.
Develop the code and generate the figures you need to solve the problems using this notebook.
For the answers that require a mathematical proof or derivation you should type them using latex. If you have never written latex before and you find it exceedingly difficult, we will likely accept handwritten solutions.
The total homework points are 100. Please note that the problems are not weighed equally.

Student details#

First Name:
Last Name:
Email:
Used generative AI to complete this assignment (Yes/No):
Which generative AI tool did you use (if applicable)?:

Problem 1 - Physics-informed Neural Networks for Solving a Neo-Hookean Hyperelasticity Problem#

*The original version of this problem was developed by Atharva Hans as a companion to this.

Consider a neo-Hookean square body defined on $(x,y) \in [0,1]^2$. Let $\mathbf{u}(x,y) = (u_1, u_2)$ describe the displacement field for this body. This body is subjected to the following displacement boundary conditions:

\[ u_1(0,y) = 0, \]

\[ u_2(0,y) = 0, \]

\[ u_1(1,y) = \delta, \]

\[ u_2(1,y) = 0, \]

with $\delta$ referring to the applied displacement along the x-direction.

For this hyperelastic material, the stored energy $E_b$ in the body can be expressed in as:

\[ E_b[\mathbf{u}(\cdot)] = \int_{[0,1]^2}\left\{\frac{1}{2}(\sum_{i=1}^2\sum_{j=1}^2{F_{ij}^2} - 2)- \ln(\det(\mathbf{F})) + 50\ln(\det(\mathbf{F}))^2\right\} dxdy, \]

with

\[ \mathbf{F} = \mathbf{I} + \nabla \mathbf{u}, \]

where $\mathbf{I}$ is an identity matrix.

The final orientation of this body is described by a displacement field that minimizes the stored energy $E_b$. The idea is to use a neural network to approximate the displacement field and train it by minimizing the stored energy $E_b$.

To automatically satisfy the boundary conditions, we will use this approximation: $$ u_1(x,y) = \delta - \delta(1-x) + x(1-x)N_1(x,y;\theta), $$ and, $$ u_2(x,y) = x(1-x)N_2(x,y;\theta) $$ where $N_1(x,y;\theta)$ and $N_2(x,y;\theta)$ are neural networks.

Part A#

Solve the problem above for $\delta=0.1$ using a physics-informed neural network (PINN). Use separate neural networks for $N_1(x,y;\theta)$ and $N_2(x,y;\theta)$. Start with a multi-layer perceptron with 3 hidden layers, each with 128 units, and tanh activations. Add a Fourier feature layer at the beginning of the network. Feel free to change the architecture if you think it is necessary.

Use equinox for the neural networks and optax for the optimization. Use a sampling average of 32 collocation points to compute the integral of the stored energy. Use the Adam optimizer with a learning rate of 0.001 for 1000 iterations to debug. Feel free to play with the learning rate, the number of collocation points, and the number of iterations.

Show the evolution of the loss function over the iterations. Plot the final displacement field (plot $u_1(x,y)$ and $u_2(x,y)$ separately).

Put your answer here. Use as many markdown and code blocks as you want.

# your code here

Part B#

Solve the problem for $\delta=0.5$ using the same architecture as above. It will likely fail to train. If yes, then use the solution of $\delta=0.1$ as the initial guess for $\delta=0.2$, and then use the solution of $\delta=0.2$ as the initial guess for $\delta=0.3$, and so on, until you reach $\delta=0.5$. This is called transfer learning.

At the end, plot the final displacement field for $\delta=0.5$.

Put your answer here. Use as many markdown and code blocks as you want.

# your code here

Part C#

Solve the parametric problem for $\delta \in [0,0.5]$. That is, build a neural network that takes $\delta$ as input and outputs the displacement field. To do this:

Modify the loss function to:

\[ \mathcal{L} = \int_0^{0.5} \int_{[0,1]^2} \left\{\frac{1}{2}(\sum_{i}\sum_{j}{F_{ij}^2} - 2)- \ln(\det(\mathbf{F})) + 50\ln(\det(\mathbf{F}))^2\right\} dxdy d\delta. \]

Modify the neural networks to take $\delta$ as input, say $N_1(x,y;\delta;\theta)$ and $N_2(x,y;\delta;\theta)$. Your field will be $\mathbf{u}(x,y;\delta;\theta)$. Use the following architecture for the neural networks:

\[ N_1(x,y;\delta) = \sum_{i=1}^n b_{1,i}(\delta)t_{1,i}(x,y). \]

Here, $n$ is your choice (start with $n=10$), $b_{1,i}$ is a neural network that takes $\delta$ as input and outputs a scalar, and $t_{1,i}(x,y)$ is a multi-layer perceptron with 3 hidden layers, each with 128 units, and tanh activations, and Fourier features at the beginning. The same applies to $N_2(x,y;\delta)$. This representation resembles an expansion in terms of basis functions. The same architecture appears in DeepONet.

Plot the $x$ and $y$ displacement at $x=0.5, y=0.5$ as a function of $\delta$.