PINN Forward Examples

This section contains examples demonstrating how to solve forward problems using Physics-Informed Neural Networks (PINNs) with physical units. These examples showcase PINNx’s unique capability to handle dimensional analysis and physical units directly in the neural network training process.

What are Forward Problems?

Forward problems involve solving partial differential equations (PDEs) where the governing equations, domain geometry, and boundary/initial conditions are fully known. The goal is to find the solution field (e.g., temperature, velocity, displacement) throughout the domain.

Physical Units in PINNx

Unlike traditional PINN implementations, these examples use PINNx’s unit-aware framework, which:

• Accepts inputs and outputs with explicit physical units (e.g., meters, seconds, Pascals)

• Automatically handles dimensional analysis during training

• Ensures physical consistency across all computations

• Improves training stability and convergence

• Makes results directly interpretable in real-world units

Featured Examples

The examples below cover a diverse range of physics problems:

• Fluid Dynamics: Beltrami flow, Burgers equation with/without adaptive refinement

• Heat Transfer: Heat equation, diffusion equation with various configurations

• Structural Mechanics: Euler beam under different loading conditions

• Wave Propagation: Helmholtz equation in various geometries

• Elliptic PDEs: Laplace equation on complex domains

Each example demonstrates best practices for:

• Setting up problems with physical units

• Defining boundary and initial conditions

• Configuring neural network architectures

• Training with dimensional awareness

• Visualizing and validating results

• Three-dimensional unsteady Navier-Stokes Equations
• One-dimensional Diffusion Equation
• Euler-Bernoulli Beam Equation
• Helmholtz equation over a 2D square domain
• Burgers equation
• Burgers equation with residual-based adaptive refinement
• Heat equation
• Heat equation with training points resampling
• Laplace equation on a disk
• Dimensional Analysis for the Laplace Equation on a Disk