Week 12 Summary: Physics-informed and constrained ML

Cross-Book Summary

1. Physics-Informed Neural Networks (PINNs)

• Embedding Laws: Enforce ODEs/PDEs via the loss function for physical consistency.
• Automatic Differentiation: Exact derivative calculations enable NNs to evaluate physical equations.
• Boundary Conditions: Methods like Lagaris substitution guarantee boundary compliance.

2. Governing Equation Discovery

• Dictionary-Based Regression: Build a dictionary of candidate math functions.
• Sparse Identification: Use regularized regression (Lasso) to discover physical laws from noisy data.
• Dimensional Reasoning: Unit analysis ensures physically plausible discoveries.

3. Constraints in Materials Science

• Monotonicity: Enforce required physical trends (e.g., hardness vs. alloying).
• Hybrid Modeling: Combine physical “White-Box” models with data-driven “Black-Boxes” (Grey-Box).

90-Minute Lecture Strategy

Part 1: Why Physics Matters

• Limits of unconstrained Black-Box models.
• Accurate but Physical models.
• PINNs need less data.

Part 2: Automatic Differentiation

• GradientTape mechanics.
• Derivatives as ML architecture components.

Part 3: Solving Physics with NNs

• PINN Architectures: Data Loss + Physics Loss.
• Enforcing Boundary Conditions.
• 3D printing heat transfer case study.

Part 4: Equation Discovery

• Sparse Regression and candidate dictionaries.
• Damped pendulum equation case study.
• Unit Analysis search pruning.

Part 5: The Grey-Box Future

• Hybrid architectures vs. FEA.
• Building industrial trust.

Quarto Website Update (Summary)

Summary for ML-PC Week 13:
- Combines neural networks with physical laws via Physics-Informed ML. - Introduces PINNs and automatic differentiation. - Details Governing Equation Discovery using sparse regression. - Applies physical constraints to build data-efficient Grey-Box models.