Inverse Physics Informed Neural Networks (I-PINNs)

Discover tensors in PyTorch, from scalars to vectors and matrices, create random and zeros tensors, perform multiplication and addition, index elements, and reshape with view and -1.

Learn how backpropagation in inverse physics informed neural networks optimizes training by forward and backward passes, using the chain rule to compute loss gradients with respect to weights.

Explore forward-time, finite-difference discretization of the Burgers equation: update u with dt, discretize du/dx and d2u/dx2, and assemble convection and diffusion terms for time stepping.

Demonstrate how the solver fails when viscosity, diffusivity, and shocks challenge the scheme, and outline using inverse physics-informed neural networks with a proper scheme to handle shocks.

Explore how physics-informed neural networks fuse neural networks with physics to solve PDEs. Train with boundary and initial conditions, enforce PDE residuals across domain points, and minimize loss.

Calculate loss by combining data loss and PDE loss using a mean squared error criterion, then backpropagate to adjust weights.

Evaluate the neural network model by changing the computational domain and grid, and examine how boundary and initial conditions influence predictions of Burgers equation using PyTorch and contour plots.

Pre-processing for solving the burgers equation with a TVD scheme, using CFL-based time stepping, initial and boundary conditions, and a Python setup to compare PINN results.

Post-process by plotting the initial and final conditions to illustrate the shock wave, and use a heat map to show diffusion and energy reduction, with a TVD-pens code comparison.

compare pins with tvd scheme using identical initial and boundary conditions and neural networks, assess convergence, and note tvd is sharper near shocks while pins remain aligned.

Create the training data for inverse PINNs by generating x–t samples with the TVD scheme, building a two-column tensor of x and time, and mapping to uf values.

Set up a neural network with a learnable lambda, and input domain data from X_train and Y_train encoded by the TVD scheme for training on the chosen device.

Declare and compute a loss function for I-PINNs by combining data loss with burgers equation PDE loss, using predictions from the model and data y_train, with a learned lambda parameter.

Define the computation domain as a rectangle with space and time domains, apply boundary conditions to U, V, and P using anchor values to guide initial conditions for PINN training.

Define the training data and loss type, then let the library compute the loss using the Navier-Stokes PDE and observable boundary conditions across domain points and boundary points.