Imagine the challenge of digitally coloring a black-and-white photo without any knowledge of the original colors. While a good algorithm might generate convincing hues, certainty about their accuracy remains elusive. This is a classic example of what’s known in science as an “ill-posed inverse problem.” Similarly, in X-ray and neutron reflectometry, researchers face a “black-and-white” scenario — reflectivity curves that support multiple interpretations and potential nanostructures. Although a single plausible solution might often be good enough to breathe life into your grandparent’s photos, in scientific inquiry — where accuracy and verifiability are paramount — relying on a single “convincing” solution is not acceptable.

*Our method helps to figure out the layers and interfaces of nanostructures (reconstructed profiles represented each by one colored curve on the right) from X-ray or neutron reflectometry curves (blue curves on the left). The bottom row (”narrow priors”) shows how our method leverages prior information to narrow down the solution among all the hypotheses compatible with the original curve*

What if instead of just searching for *the* single best solution, we ask how probable is *any* given solution in our case (i.e. given the measured data)? This question reframes the problem in a probabilistic manner. Indeed, this approach is significantly more challenging than searching for a single solution. Remarkably, modern deep learning methods can explore all possible solutions efficiently and reliably!

Let's return to the black-and-white photo analogy: imagine you remember the color of your grandmother's hat in an old photograph. Incorporating this bit of prior knowledge can significantly narrow down the range of potential colorizations the algorithm might propose. In our recent work, we introduce a method that leverages just this type of insight — employing prior knowledge to dramatically reduce ambiguity in processing reflectometry data, ensuring our interpretations are both precise and plausible.

*This work is done is a collaboration with Dr. Alexander Hinderhofer, Dr. Alexander Gerlach, and Prof. Dr. Frank Schreiber from **the University of Tübingen** and **Maximilian Dax from the MPI-IS**. *