Bayesian optimization
Niobia recommends the next experiment to run.
Bayesian optimization picks the next experiment to run. It builds a surrogate model of the response with its uncertainty, then balances exploiting the high-predicted regions against exploring the uncertain ones, choosing the run at the peak of an acquisition function. It is how you reach an optimum in the fewest experiments.
What it measures
Bayesian optimization is a closed loop for expensive experiments, where each run costs real time and material and you cannot afford a dense grid:
- The surrogate: a model, usually a Gaussian process, fit to the runs so far. It gives both a predicted response everywhere and an uncertainty band that is tight near sampled points and wide in unexplored regions.
- The acquisition function: a rule that scores every candidate next run by combining the prediction and the uncertainty, expected improvement, upper confidence bound, or probability of improvement. Its peak is the most valuable next experiment.
- The explore-exploit balance: exploiting means sampling where the predicted response is already high; exploring means sampling where the model is uncertain and a surprise could hide. The acquisition function trades these off automatically, so the search neither tunnels into a local peak nor wanders aimlessly.
Run the chosen experiment, add the result, refit the surrogate, and repeat. Each iteration spends one run to learn the most it can about where the optimum is.
How to read the output
Watch the surrogate and the acquisition function together. Early on, the model is uncertain and the acquisition function sends runs into unexplored regions, exploration dominates. As evidence accumulates, the uncertainty collapses where you have sampled and the acquisition peak migrates toward the genuine optimum, exploitation takes over. A healthy run shows that progression: scattered early runs, then convergence. The judgment calls are the acquisition function (expected improvement is a sensible default; upper confidence bound lets you dial exploration up or down) and knowing when to stop, when the best expected improvement of the next run has dropped below what another experiment is worth. The method's whole promise is fewer runs to the optimum, so the stopping decision is part of the value.
A real use case
An electrolyte formulation team is optimizing a four-additive blend for cycle life, and each candidate means building and cycling cells for weeks, so a grid search is out of the question. Bayesian optimization starts from a small space-filling Latin hypercube set, fits a Gaussian process to the cycle-life results, and recommends the next blend at the acquisition peak, sometimes a blend predicted to be good, sometimes one the model is simply unsure about. Over a dozen iterations it converges on a blend well outside the team's intuition, in a fraction of the experiments a factorial sweep of four additives would have demanded. The agent recommending the next run, rather than a human guessing, is what compresses months of formulation work into weeks.
Common mistakes
- Pure exploitation: always sampling the predicted best and never the uncertain regions, which locks onto a local optimum and misses a better one elsewhere.
- Starting with too few initial points, so the surrogate is badly wrong and the early acquisition recommendations are noise.
- Ignoring the surrogate's uncertainty, the part that makes the explore-exploit balance possible. A model that does not know where it is ignorant cannot guide the search.
- Running it on a cheap, fast response where a dense grid or a factorial would simply be better. Bayesian optimization earns its complexity when each run is expensive.
- Never stopping. The point is fewer runs; continuing past the point where expected improvement is negligible spends experiments for nothing.
Niobia recommends the next experiment
Niobia runs the Bayesian optimization loop: it fits a Gaussian process surrogate to the experiments so far, evaluates the acquisition function across the factor space, and recommends the next run at its peak, balancing exploiting high predicted response against exploring high uncertainty. After each result it refits and recommends again, so the search converges on the optimum in as few runs as the problem allows. It seeds the loop with a space-filling Latin hypercube sample, and the same surrogate can be explored interactively in a prediction profiler. This is the predictive end of the DOE workflow: not just analyzing a fixed design, but choosing what to run next.
Frequently asked
How is Bayesian optimization different from a classical DOE?
A classical DOE fixes all the runs up front from a design; Bayesian optimization is adaptive, it chooses each next run based on everything learned so far. That makes it far more efficient at finding an optimum when experiments are expensive, because it spends each run where it will learn the most.
What is the acquisition function?
The rule that decides the next run by combining the surrogate's prediction and its uncertainty. Expected improvement, upper confidence bound, and probability of improvement are common choices. Its peak is the experiment that best balances exploiting known-good regions against exploring uncertain ones.
When is it the wrong tool?
When experiments are cheap and fast, a dense grid or a factorial design is simpler and just as good. Bayesian optimization earns its complexity when each run costs significant time or material, which is exactly the regime of cell builds and long cycling tests.