PINN Forward Unitless Examples

PINN Forward Unitless Examples#

This section contains examples demonstrating how to solve forward problems using Physics-Informed Neural Networks (PINNs) without explicit physical units. These examples work with dimensionless or normalized equations, following the traditional PINN approach.

What are Unitless Examples?#

Unitless (or dimensionless) examples use normalized variables and equations where:

  • All quantities are scaled to be of order unity (O(1))

  • Physical dimensions are removed through non-dimensionalization

  • Equations are expressed in terms of dimensionless parameters (e.g., Reynolds number, Mach number)

  • Results require post-processing to convert back to physical units

This approach is widely used in traditional computational physics and can be beneficial when:

  • Working with multi-scale problems

  • Comparing solutions across different parameter regimes

  • Simplifying complex equations

  • Following established non-dimensional formulations

Comparison with Unit-Aware Examples#

While PINNx supports both unit-aware and unitless formulations, each has its advantages:

Unitless (this section)

Unit-Aware (see PINN Forward Examples)

Traditional approach

PINNx’s innovative feature

Requires manual non-dimensionalization

Automatic dimensional handling

Results need scaling back

Results in physical units directly

Better for classical benchmarks

Better for real-world applications

More examples available

Growing collection

Comprehensive Example Collection#

This section features an extensive collection of over 30 examples covering:

Elliptic PDEs#

  • Poisson Equation: Various boundary conditions (Dirichlet, Neumann, Robin, Periodic)

  • Helmholtz Equation: Multiple geometries including domains with holes

  • Laplace Equation: Complex domain shapes

Parabolic PDEs#

  • Diffusion Equation: Standard, with exact BC, with resampling

  • Heat Equation: Time-dependent problems

  • Diffusion-Reaction: Coupled equations

Hyperbolic PDEs#

  • Klein-Gordon Equation: Wave propagation

  • Schrodinger Equation: Quantum mechanics

Nonlinear PDEs#

  • Allen-Cahn Equation: Phase field modeling

  • Burgers Equation: Nonlinear advection-diffusion

  • Navier-Stokes: Fluid dynamics (Kovasznay flow, Beltrami flow)

Fractional PDEs#

  • Fractional Poisson: 1D, 2D, and 3D formulations

  • Fractional Diffusion: Time-fractional derivatives

Structural Mechanics#

  • Euler Beam: Classical beam theory

  • Linear Elasticity: 2D plate problems

Ordinary Differential Equations (ODEs)#

  • Second-order ODEs: Various boundary conditions

  • ODE Systems: Coupled equations

  • Lotka-Volterra: Population dynamics

  • Volterra Integro-Differential Equations (IDE): Memory effects

Advanced Features Demonstrated#

  • Hyperparameter Optimization (HPO): Automated tuning

  • Hard Constraints: Exact boundary condition enforcement

  • Residual-based Adaptive Refinement (RAR): Dynamic point resampling

  • Point Set Operators: Custom boundary operators

Each example includes detailed implementations showing:

  • Problem formulation and non-dimensionalization

  • Neural network architecture selection

  • Training configuration and loss functions

  • Result visualization and validation

  • Comparison with analytical/numerical solutions (when available)