Approximating the 1D wave equation using Physics Informed Neural Networks (PINNs)

An implementation of Physics-Informed Neural Networks (PINNs) to solve various forward and inverse problems for the 1 dimensional wave equation.

Detailed explanation

Wave_equation.py solves the 1d wave equation

Wave_equation_otherBC solves the 1d wave equation with Neumann boundary conditions

test_loss_time.py shows the test error-computational time dependency for a specific structure of neural network

train_error_val_error_time.py displays the train error/validation error/computational time-size of training set dependencies

all_together_loss_time.py shows the test error-computational time dependency for different structures of neural networks.

changing_nodes_test_loss.py shows the test error-computational time dependency for neural network structures with different numbers of nodes.

inverse_problem.py solves the inverse problem of the 1d wave equation

first_case_no_damage.py solves the degenerating 1d wave equation when $a(x)=4$

second_case_damage.py solves the degenerating 1d wave equation when $a(x)=8|x-0.5|$

third_case_double_damage.py solves the degenerating 1d wave equation when $a(x)=16|x-0.5|^2$

control.py solves the null controllability problem of the 1d wave equation.

degenerate_wave.m solves a wave equation $u_{tt}(t,x) + a(x) u_{xx}(t,x) = 0$ on $x \in (0,1)$ in which the stiffness is $a(x) = 4(2|x-\frac{1}{2}|)^\alpha$ by finite differences. (used for comparison)