Traditional ML: Learn patterns from data alone

PINNs: Learn patterns from data AND physics simultaneously

Key Advantages of PINNs

1. Regularization Effect: Physics constraints prevent overfitting
   • Standard NN can fit any function through the data points
   • PINN is constrained to solutions that satisfy the ODE
2. Better Interpolation: Smooth, physically meaningful predictions between data points
   • Standard NN: arbitrary interpolation
   • PINN: physics-guided interpolation
3. Accurate Derivatives: Natural consequence of physics enforcement
   • Automatic differentiation + physics loss = correct derivatives
   • Critical for applications requiring gradients (optimization, control)
4. Data Efficiency: Less training data needed
   • Physics provides strong inductive bias
   • Can generalize from very sparse measurements

The Universal Approximation Foundation

Why this works theoretically:

• Neural networks can approximate functions in Sobolev spaces \( H^k \)
• Sobolev spaces include both functions AND their derivatives
• With smooth activation functions (\( \tanh \), \( \sin \)), we can approximate solutions to differential equations
• Automatic differentiation makes this practical

When to Use PINNs

Ideal scenarios:

• ✅ Known governing equations (PDEs/ODEs)
• ✅ Sparse, noisy data
• ✅ Need physically consistent solutions
• ✅ Require accurate derivatives
• ✅ Complex geometries (where finite elements struggle)

Limitations:

• ❌ Unknown physics
• ❌ Highly nonlinear/chaotic systems
• ❌ Large-scale problems (computational cost)
• ❌ Discontinuous solutions

Extensions and Applications

This framework extends to:

• Partial Differential Equations: Heat equation, wave equation, Navier-Stokes
• Inverse Problems: Estimate unknown parameters from data
• Multi-physics: Coupled systems (fluid-structure interaction)
• High Dimensions: Curse of dimensionality breaking

The Bottom Line

PINNs = Universal Function Approximation + Physics Constraints + Automatic Differentiation

This combination creates a powerful method for solving differential equations with neural networks, particularly when data is sparse and physics is well-understood.

Next Steps: Try this approach on the 1D Poisson equation with hard constraints!