Machine Learning in Materials Processing & Characterization
Unit 9: Inverse Problems and Process Maps

Prof. Dr. Philipp Pelz

FAU Erlangen-Nürnberg

FAU Logo IMN Logo CENEM Logo ERC Logo Eclipse Logo

0. The Engineering Dream

From Prediction to Design

Today’s Learning Journey:

  • Forward vs. Inverse: Why design is harder than prediction.
  • PIML: Injecting physics into the learning process.
  • PINNs: Physics-Informed Neural Networks.
  • Automatic Differentiation: The engine of scientific ML.
  • Operator Learning: Learning the laws, not just the solutions.

1. Forward vs. Inverse Problems

The Direction of Discovery

  • Forward: Processing \(\to\) Property. (Well-posed).
  • Inverse: Property \(\to\) Processing. (Ill-posed).

Challenges of Inversion

  • Non-uniqueness: Multiple paths to the same property.
  • Instability: Small changes in targets \(\to\) Massive changes in recipes.

The Solution: Using physics as a “regularizer” to guide the model toward valid solutions.

2. Physics-Informed Machine Learning

Why hide information?

“Why hide knowledge from an algorithm when it would improve training?” Neuer, Michael et al., (2024)

Three Types of Bias

  1. Observational: Train on simulation data.
  2. Learning: Embed laws in the loss function.
  3. Inductive: Specialized architectures (e.g. Symmetries).

3. PINNs

Physics-Informed Neural Networks

  • A simple MLP that satisfies Partial Differential Equations (PDEs). Sandfeld, Stefan et al., (2024)
  • The Total Loss: \[L = w_{data} L_{data} + w_{phys} L_{residual}\]
  • Residuals: Forcing the output to follow the Heat Equation, Navier-Stokes, or Diffusion laws.

4. Automatic Differentiation (AD)

The Engine of scientific ML

  • To solve \(L_{residual}\), we need derivatives (e.g. \(\frac{\partial T}{\partial t}\)).
  • AD: Exact gradients computed via the chain rule. Neuer, Michael et al., (2024)
  • Unlike numerical differentiation, AD is precise and scales to thousands of parameters.

5. Operator Learning

DeepONet

  • Instead of learning a single solution, we learn the Operator (the law itself). Sandfeld, Stefan et al., (2024)
  • Branch Net: Processes input functions (e.g. sensor logs).
  • Trunk Net: Processes spatial coordinates.
  • Result: Instantaneous physical predictions for new boundary conditions.

6. Summary & Takeaways

Key Messages:

  1. Inverse Design transforms ML from a “prediction tool” to a “design tool.”
  2. PIML enables training on small, sparse experimental datasets.
  3. PINNs provide physical grounding and explainability.
  4. Operator Learning (DeepONet) is the future of real-time industrial control.

Exercise Handoff

  • Implement a Custom Loss in PyTorch for a 1D heat equation.
  • Use GradientTape to compute the residuals.
  • Discover the unknown thermal diffusivity \(\alpha\) from noisy temperature logs.

References

Machine learning for engineers: Introduction to physics-informed, explainable learning methods for AI in engineering applications, Michael Neuer & others
Materials data science, Stefan Sandfeld & others