Inverse problems and differentiable design
A temperature field can reveal a heat source. A decaying signal can reveal a time scale. But different materials and different sources can also produce the same observations. This book teaches you to ask both questions: which parameters fit, and what have the measurements actually told us?
We begin with a decay rate you can recover on paper. Then we build a steady Poisson model, recover a source amplitude, and follow derivatives through its solution. The final investigation chooses measurements and parameter scales that make the inference informative.
Choose your starting point
Section titled “Choose your starting point”| Investigation | Question you will answer | Useful starting knowledge |
|---|---|---|
| 1. Measurements to parameters | What can one decay measurement tell me? | Exponentials and logarithms |
| 2. Recovering a heat source | Can a field reveal the strength of its source? | Gradients and steady balance, recapped locally |
| 3. Sensitivity and identifiability | Why can a perfect fit leave a parameter unknown? | Partial derivatives |
| 4. Differentiating a solved model | How does a source change propagate through a solver? | Matrix multiplication and the chain rule |
| 5. Noise, scaling, and design | Which measurements should I take next? | The recovery formula from chapter 2 |
Read in order for a connected course, or enter through a question that interests you. Each chapter contains a worked calculation, an experiment or prediction, and exercises. You can follow the mathematical argument without installing software. Use Get started for the shared Python environment when you want to run the examples.
One physical thread, several perspectives
Section titled “One physical thread, several perspectives”The running field problem is a square held at a fixed boundary temperature and heated smoothly inside. The heat-transfer book explains energy balance and conduction. Here we ask what the observed response says about the source and conductivity. The numerical-simulation book explains why a computed field changes when its mesh changes.
By the end, you will have an analytic parameter estimate, an Eqiora field calculation, a sensitivity experiment that exposes an ambiguity, and a reasoned choice of additional observations. These are the ingredients needed before a large optimization problem becomes useful.