Physics Informed Neural Networks - Burgers' Equation

The viscous Burgers' equation in one spatial dimension is given by $$\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \nu\frac{\partial^2 u}{\partial x^2}.$$

In this project we use a Physics Informed Neural Network (PINN) to find the solution to this equation. Here we minimise the loss functional without any training data (unsupervised learning). We construct the loss function using the PDE described above and the boundary conditions $u(x,t=0) = -\text{sin}(x)$ and $u(x=0,t) = u(x=1,t) = 0$ as

$$\mathcal{L}(\theta) = \frac{1}{N}\sum^N\left(\frac{\partial NN}{\partial t} + u\frac{\partial NN}{\partial x} - \nu\frac{\partial^2 NN}{\partial x^2}\right)^2 + \frac{1}{N}\sum^N(NN(\theta;x,t=0)+\text{sin}(x))^2 + \frac{1}{N}\sum^N(NN(\theta;x=0;t))^2- + \frac{1}{N}\sum^N(NN(\theta;x=1;t))^2$$.

The rest using PyTorch to train the PINN. The PyTorch autograd.grad() function is helpful for calculating the spatial and temporal gradients of the PINN output predicted fields. I am still trying to figure out the best parameters to fit the model.