Physics-Informed-Neural-Networks

• Bachelor Thesis PDF
• Presentation PDF

Bachelor Thesis on Physics Informed Neural Networks for Identification and Forecasting of Chaotic Dynamics. We investigate the ability of physics informed neural networks — data-driven neural networks which incorporate laws of physics generally in the form of partial differential equations (PDEs) — to infer the solutions of and to identify a number of nonlinear partial differential equations. A relatively simple physics informed neural network framework was created that can be easily adjusted and applied to solve any of the mentioned problems concerning different partial differential equations.

Code

Each of the jupyter notebooks is an end to end application of one of our approaches for different PDEs.

Burger's Equation

Being a 2nd order PDE, the Burger's equation was a perferct benchmark example for our appraoches

Burgers_Continuous_Inference.ipynb

Given PDE parameters and low amounts of boundary and initial data, this framework infers the spatio-temporal behaviour of the Burger's equaiton in cotious time.

Burgers_Discrete_Inference.ipynb

Given PDE parameters and low amounts of boundary and initial data , this framework infers the spatio-temporal behaviour of the Burger's equaiton in discrete time.

Burgers_Continuous_Identification.ipynb

Given spatio-temporal data, this framework infers the PDE parameters of the Burger's equaiton in continuous time.

Burgers_Discrete_Identification.ipynb

Given spatio-temporal data, this framework infers the PDE parameters of the Burger's equaiton in discrete time.

Nonlinear Schrödinger equation

The Nonlinear Schrödinger equation, a popular equation in theoretical physics, presented us with a new challange: its complex output required our approach to infer a 2D output as opposed to the example in the Burger's equation which was only 1D.

Schrodinger_Continuous.ipynb

Given PDE parameters and low amounts of boundary and initial data, this framework infers the spatio-temporal behaviour of the Nonlinear Schrödinger equaiton in cotious time.

Kuramoto-Sivashinsky Equation

The Kuramoto-Sivashinsky equation is a 4th order PDE used to model instabilities in a laminar flame front. Modelling the Kuramoto-Sivashinsky equation is a difficult task due to its chaotic nature.

Kuramoto_Sivashinsky_Continuous_Time_Identification.ipynb

Given spatio-temporal data, this framework infers the PDE parameters of the Kuramoto-Sivashinsky equaiton in continuous time.

Kuramoto_sivashinsky_discrete_identification.ipynb

Given spatio-temporal data, this framework infers the PDE parameters of the Kuramoto-Sivashinsky equaiton in discrete time.