Practical Bayesian Optimization in the Fab (2026-US-30MP-2920)

Thank you for joining me for this session. We're going to be talking a little bit about Bayesian optimization, specifically in the context of the semiconductor fab environment. But there's lots of applications on this topic in various high-volume manufacturing contexts. There are three main points that I want you to get from this session, and we'll go through them right here so you can see them throughout… Carry that thread throughout the session.

The first is a basic baseline statement. When we're talking about experimentation in a high-volume manufacturing environment like a semiconductor fab, definitive screening designs are a must-use. For those of us in the fab, the concept of being able to put 10 experimental factors with center points and axial edge points into a single FOUP, into 25 experiments, into 25 wafers, that's a very appealing idea for us. That's something that we want to keep in mind.

The other points are what we're going to build on from that foundation is that this idea that Bayesian optimization can provide some enhanced flexibility, particularly in the resource-constrained environments like the fab. It gives us the ability to explore and optimize simultaneously. It gives us the ability to explore some of the complex surfaces that we're used to seeing in the fab environment that are sometimes difficult to capture with traditional quadratic-type models that we tend to run in the fab.

The second is that when we combine… The last is that when we combine definitive screening designs with Bayesian optimization, we enhance the ability of Bayesian optimization, both in a business process context, and it turns out just from a pure math and efficiency context as well. We get better use of resources, we get to solutions faster, and we tend to get better quality solutions. That's what we're going to hope to learn today.

Let's take a step back and talk about the problem that we're going to be dealing with. I've already mentioned that resource constraint that we've got. Twenty-five wafers are in a lot. It's very hard to get more than one lot. If you're lucky, you can get one lot of material, 25 wafers. When that happens, we can also introduce problems, blocking issues, lot-to-lot variation, wafer-to-wafer variation, breakage, all kinds of those things come into the equation that make the concept of experimentation in a fab a little bit more challenging.

The one that people don't talk about is this business process cost. Every time we want to go a step down our experimentation path, we generally have to check in with a change review board of some kind. How quickly we can get in front of that board, how quickly they can make a determination on the value of what we're doing and whether or not they'll apply approved material, that adds a different cadence on top of the actual fab learning cadence, the move rate that goes through the fab itself.

Those two together make this idea that large-scale experimentation, this iteration, when we take a DOE class, we talk about the screen, model, refine, validate, that flow breaks down in the fab. It becomes really hard to follow that classic DOE strategy that we've been taught and that works in R&D environments and things like that. It really doesn't overlay very nicely when you're trying to experiment in a production facility.

That brings us to where we're going to be talking about four different questions. First, for the uninitiated, I'm going to do a quick introduction into Bayesian optimization. Then I'm going to do a quick introduction into definitive screening designs, and then we'll talk about what happens when we mix the two. I want to show you a meta-study that I did where I went through and pulled out a bunch of papers where they published the data and they published the models, and reanalyzed the experiments to see what would have happened if we had run them using this new strategy.

Okay, question one, what's Bayesian optimization? It's an iterative method of modeling and augmenting existing datasets. That's an important point to note. It doesn't generate data on its own out of the gate, you have to provide it with something. That initialization dataset is really critical. Behind the scenes, it uses a Gaussian process model for the modeling component. That's what allows it to do some fanciness with the surfaces that it's looking at. It uses this thing called an expected improvement function to figure out where the next experiments it's going to recommend are going to be. That's what it does.

Now, let's look at an example here just to help you understand where all that happens. This surface, this three-peak surface that we've got, I've got the initialization dataset here. Those are the purple data points that we have. When I fed this into the Bayesian optimization tool, it said, "Hey, I don't have enough information. I need to do some exploratory analysis." That's that expected improvement function saying, "Hey, we need some more information." It went through and it did four rounds of space filling, trying to figure out where it was in space, those green points.

It turns out at that point it says, "Hey, I can now start optimizing." Then it goes through 13-odd rounds of optimization. First, it started over here on this little peak, and then it went, "Oh, that's not good," and started climbing up this peak right here until it ended up at the top. Then when it ran out of room at the top, it just started replicating at the top and saying, "Hey, that's my best. That's my best." That's the process of Bayesian optimization. We give it a dataset, it analyzes it, says, "Hey, I don't have enough information," or says, "Hey, I do have enough information," and starts going through and iterating on the goal that we've set for it. In this case, it was to find a peak.

There are a couple of things that are really critical that I want to call out from that little exercise that are going to govern what happens going forward. The first is there is this borderline marketing speak that you can put anything into Bayesian optimization and it will get a toehold and start optimizing. That is 100% true, but it will take a while. That's the part that commonly gets left out.

The thing that most of the literature, and that I'm going to demonstrate here, is that a planned strategy, a planned initialization dataset, like a definitive screening design or Latin hypercube or any of these other design strategies that are out there, is required for peak performance for this algorithm to work at its absolute best. You can put incomplete data into it, absolutely, it's going to take a moment and it's going to do some space filling to fill in the gaps. That's the first thing we need to be prepared of.

The second thing is that within the planned initialization datasets, there's actually a hierarchy. There are different qualities of those planned datasets, and the quality of the initialization dataset will actually influence further how quickly the algorithm is able to converge to its global optimum. Planned versus not planned, you always want to have planned. Within planned, there's actually good, better, best for planned initialization set strategies. That's where we lead into definitive screening designs because DOE is considered the gold standard for planned experimentation, and definitive screening designs have unique properties that are really desirable for us in the fab.

What does it have? They have very high orthogonality, that's the first thing we want to remember. They've got… And I'll show you what that looks like in a moment, they have center and axial edge points. They don't scale the same way that classical designs do. In the modern implementation, they scale as 2 to the k plus 5 factors for k factors. I did a little math over here to show you what that looks like. A full factorial, the number of experiments scales as 2 to the k. For 10 factors, that's going to take 1,024 runs. Not reasonable.

We have this other thing called a fractional factorial design, where we reduce the ability of the experiment to capture more nuanced effects. We give up that information in order to bring the number of runs down. For a third order or for a fractionation of three, we can bring this 1,024 number down to 128. Optimal RSM, that's another one we talk about. That one we wouldn't even try with 10 factors. With six, we can get that number down to 77. For a DSD, for definitive screening design, we can get 10 factors into 25 experiments. That's that golden number of wafers that we deal with in the fab. That's a lot, a lot of material.

What do these look like? If you look at them in projection in any one pair of dimensions, they look like a standard response surface model, which is pretty powerful. That says we get these actual axial edge points here that I'm talking about, we get center points as well. Actually, we get multiple center points. Just looking at three of these factors, you can see that when you look at it in 3D, that we're giving up a few points that are less informative, a little bit more redundant, in exchange for some additional efficiency. That's more or less what we're doing.

Now, the question that I wanted to look at when I started this study was what happens if we use DSDs as the training sets, as those initialization sets for Bayesian optimization? I went and ran a bunch of simulations. I used some standard benchmarking functions. These are known for testing the ability and efficiency of optimization algorithms. The first one is an offset sphere, so just standard spherical function, just offset off of the 0, 0 origin. The rest of these have various little gotchas that make it difficult for optimization functions or optimization algorithms to find the global minima.

These all have different gotchas. The Rosenbrock is considered one of the gold standards. It's really quite difficult for functions to find this little minima that's right here. What I did was I set up a simulation where I let it have four factors, five factors, 10 factors, 20 factors like that. Then I looked at how fast the DSD, definitive screening design, a Latin hypercube, and just a random space filling strategy behaved. The results were really interesting. For the sphere… We'll move this over here. The one we want to pay attention to, the yellow line is the DSD, the Latin hypercube, that's the gold standard in the literature for this, is the blue, and just a pure random scattershot within the design space is the third one.

For a DSD, we can see that it performs quite well on its own. The other thing to note is that there's a different scaling strategy for Latin hypercube and random. They tend to scale as three times the number of factors. I allowed them to do that in this set of simulations, and then I constrained them to be apples to apples, same number of starts as… Or same number of experiments as a definitive screening design over here, just to make sure we were comparing everything.

The results really aren't that different. This we can see things scale incredibly quickly with these, whereas the DSD tends to stay nice and flat. Over when we have the same number of starts, there's a little bit more overlap, but largely the definitive screening design is still beating out when it comes to the spheres. For the Ackley, story is pretty much the same. It's actually quite a lot more distinct. We can see a lot more daylight between the two or between the three strategies.

The Rastrigin, let me move this out of the way so you can see the Rastrigin there. That's this little guy right here, this bed of pincushions or bed of pins. It gets more interesting. There's a lot more overlap when we are dealing with the same number of starts, when we have the same size of initialization set. Then the Rosenbrock, it's pretty much the same. By the time we hit 20 factors, everything converges there, which is more or less what you'd expect. But the point that we want to take from this is that consistently across these different benchmark simulations or these different simulation benchmark functions, the DSD is beating out the best in class every single time, or at least doing as well, which is quite an interesting finding.

That brought me to this next question. What would happen if we put this in the fab? Well, I don't happen to have a fab in my backyard. I went through and looked back through the literature and found a number of studies where they had published their data, they had published the models, and they basically let me do a rerun of their studies. These are the three that I found… That I wanted to look at. They're different numbers of factors, all within reason. These are all reasonable numbers of factors.

The first is a dry etch study where we looked at a sequential designed experiment where we used a sequential DSD, so definitive screening design, followed by sequential experimentation versus a DSD and a Bayes Opt strategy. The next was I wanted to explore that Latin hypercube versus definitive screening design in a little more detail. Then the last, this is what I'm going to actually show a little bit more detail on directly, is this just going, what would happen if I just ran a definitive screening design versus a definitive screening design and then added a little bit more optimization using a Bayesian optimization?

I'll give you the story so you can see what they look like in general. The dry etch, they got the exact same number of experiments within one. That's within error. That's not that big of a deal, that one extra experiment. DSD did take an extra learning cycle, but it did have an improvement in uniformity. It found an optimized solution that was not possible with the sequential experimentation strategy.

When it came to the Latin hypercube versus definitive screening design, it was a little ugly. The Latin hypercube took 18 rounds of experimentation to get to the same optimum that the definitive screening design-based Bayesian optimization got in five, so 72% reduction in the learning cycles between the two. Then looking at what we get when we just bolt on Bayesian optimization to a definitive screening design, we do have to iterate, so it's going to lose on the number of runs. But we did actually get a 5.5% improvement rate. We found a better solution than what was published in the literature.

That's important because that means we have essentially built out additional capacity with our existing installs. There's this strategy in many semiconductor companies, more out of existing install, that's essentially what we do. We found 5% more capacity out of an existing tool, which is free money.

Let's have a little bit of a look at this a little bit more closely. Here's the definitive… Here's this case study. You can see the blue dots or the blue triangles, that's the DSD laid over. You can see where the Bayesian optimization immediately started gravitating towards specific parts of the design space. Now, looking at design spaces in six dimensions is tricky. One of the strategies that we use for monitoring how Bayesian optimization pays attention to things is by using principal components analysis.

You do a principal components analysis on the factors and you plot the first two factors. That's the majority of the variability, the majority of what the algorithm is moving in order to explore the design space is captured in those first two factors. That's what this looks like. I've got a better view for you as well. The solution that was found in the literature is right here at a design point, the darker the blue, the more desirable it is. It's found at a reasonably good point. But when we ran a few iterations of Bayesian optimization on top of it, it turns out that there's a better solution just a little bit further away. That's the important thing to note.

In fact, you can see it quite clearly when we look at it in 3D. If we look at the desirability, this is principal component one, principal component two. There's where the definitive screening design went initially, and it went up here. Then there's the point that the literature found. Then over here is the point that had that slightly better optimum. The other thing we can see is that there's probably, if we wanted to take this a little further, there's probably an even better solution a little bit further on, but the dataset itself being true to what was in the published data, I didn't want to go outside of what was available to me.

But there's good evidence to say that there's probably another solution out here that's even better, which is something we could explore fairly easily with Bayesian. Again, we can either hit as good a value as what we found, we can get there a little faster, or we can generally get a better result. Let's circle back to those three thoughts that I wanted to make sure that we got. One, I hope you've seen that definitive screening designs in general are something we need to be paying attention to in the fab. The efficiency gains are just something we can't ignore.

When we tack on Bayesian optimization to it, we're going to find solutions that we might not expect that are probably better than what we could get with just DOE alone. Likely we'll get there a little faster. The last thing, again, is that all of this was constrained within the business model that we have to work with, within that change review board that we have to consider.

We can explore and optimize simultaneously, which allows us to go to the board once and say, "Hey, these are the experiments we want to run. This one's going to be looking for the optimum, these are going to be helping us flush it out and make sure that we can know what's going on and be ready for the next set of experiments."