Physics_Informed_Neural_Networks

This project demonstrates how PINNs can be used as a mesh-free alternative for solving elasticity problems while preserving physical consistency through the loss formulation.

Problem Setup

This project studies the simple bending of a 2D square domain under plane stress conditions. The right edge is fixed, while a prescribed shear traction is applied on the left edge:

$$ \sigma_{xy} = t_y, \qquad t_y = 0.01 ,\text{N/m}^2 $$

The objective is to use a Physics-Informed Neural Network (PINN) to approximate the displacement and stress fields of the domain.

Methodology

The spatial coordinates $(x, y)$ are generated randomly:

• inside the square domain,
• and along its boundary.

These coordinates are used as the inputs to the neural network.

The network predicts the following output fields:

$$ u_x,; u_y,; \sigma_{xx},; \sigma_{yy},; \sigma_{xy} $$

To enforce the physical behavior of the problem, the model uses:

• equilibrium equations,
• strain-displacement relations,
• and plane stress constitutive equations.

Using the predicted displacement field, the strain components

$$ \varepsilon_{xx},; \varepsilon_{yy},; \varepsilon_{xy} $$

are computed, and from these, the corresponding stress relations are evaluated.

Loss Function

The PINN is trained by minimizing residuals derived from the governing equations and boundary conditions.

The residuals are obtained by comparing:

• the stress components predicted by the neural network, and
• the stress components computed from the constitutive relations.

Mean Squared Error (MSE) losses are then defined for:

• equilibrium equations,
• displacement constraints,
• additional boundary conditions,
• prescribed boundary conditions.

The total loss is written as:

$$ \mathrm{Loss}_{\mathrm{total}} = \mathrm{Loss}_{\mathrm{balance},x,y} + \mathrm{Loss}_{\mathrm{displacement},x,y} + \mathrm{Loss}_{\mathrm{additional\ BC}} + \mathrm{Loss}_{\mathrm{prescribed\ BC}} $$

Test Cases

The following test case is considered:

Case 1:
The weight factors in the individual loss terms are varied, and the resulting model performance is compared. The best configuration is selected based on the lowest absolute and relative errors in the predicted stress components.

Goal of the Project

The purpose of this project is to investigate how effectively a PINN can solve a 2D elasticity bending problem and how the choice of loss weighting influences the accuracy of the predicted displacement and stress fields.