By, SK. Aftab Hosen ([email protected]) , Bitan Mukherjee ( [email protected]) , Srijita Bhattacharyya ( [email protected]) ; Jadavpur University, Kolkata, WB
In this article, we explore how to solve the lid-driven cavity problem using Physics-Informed Neural Networks (PINNs). PINNs integrate the governing physical laws, expressed as partial differential equations (PDEs), into the training process of neural networks. This approach allows us to approximate the solution to PDEs without the need for traditional discretization methods like finite elements or finite volumes.
Physics-Informed Neural Networks (PINNs) are a class of neural networks that incorporate physical laws into the learning process. Instead of solely relying on data, PINNs embed the governing equations (e.g., Navier-Stokes equations for fluid flow) into the loss function. This means that the network is trained not just to fit data but also to satisfy the underlying physics.
We aim to determine the temperature profile $T(x,y,t)$ of a metal plate over time using Physics-Informed Neural Networks (PINNs). The metal plate is modeled as a two-dimensional domain $(x,y)∈[0,1]×[0,1]$ over a simulation time interval $t∈[0,40]$ seconds. The heat conduction within the plate is governed by the transient heat equation. One side of the plate $(x=0, y=0)$ is continuously heated to a fixed temperature of $T_{\text{heated}} = 100^\circ\text{C}$ while the rest of the boundaries are insulated (i.e., no heat flux crosses these boundaries). Initially, the plate is uniformly at $T_{\text{initial}} = 20^\circ\text{C}.$
The heat conduction in the plate is described by the 2D transient heat equation:
$$ ∂T/∂t = \alpha \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} \right) $$
where the thermal diffusivity α is defined by
$α= \frac{k}{\rho C_p}$
with: