Case study · Searching 800,000 shape memory alloys with expected improvement

Case study · Searching 800,000 shape memory alloys with expected improvement#

At a glance

Why this matters#

Alloy design poses the same problem as the reactor in L15: every evaluation is expensive, the budget is a few dozen experiments, and there are far too many candidates to test. The reactor has one or two knobs. An alloy with three substituted elements, each adjustable in steps of 0.1%, has hundreds of thousands of candidate compositions, and each one has to be melted, rolled, heat treated and measured.

Xue and colleagues at Los Alamos National Laboratory and Xi’an Jiaotong University ran that search for nickel-titanium shape memory alloys with low thermal hysteresis. They started from 22 measured alloys, used a regression model with an uncertainty estimate to propose four new compositions at a time, synthesized them, and repeated. After nine rounds and 36 new alloys, they had found 14 alloys better than anything in their starting data. The paper is open access, and its design choices map closely onto the vocabulary of L14 and L15.

The setting#

A shape memory alloy changes crystal structure when it is heated or cooled through a transformation temperature, between a high-temperature austenite phase and a low-temperature martensite phase. The transformation on heating and the one on cooling do not happen at the same temperature. The gap is the thermal hysteresis, written \(\Delta T\). The paper links hysteresis to fatigue, citing earlier work in which 60 heating and cooling cycles shifted the transformation temperature of Ni\(_{50}\)Ti\(_{50}\) by 25 K. The design goal was the composition with the smallest \(\Delta T\), so this is a minimization problem.

The authors restricted the search to one family, Ni\(_{50-x-y-z}\)Ti\(_{50}\)Cu\(_x\)Fe\(_y\)Pd\(_z\). Copper, iron and palladium replace part of the nickel, each in steps of 0.1%, subject to limits on each element (at least 30% nickel, at most 20% copper, 5% iron and 20% palladium). The paper counts \(N = 797{,}504\) possible alloys in this space. The group had synthesized 22 of them under identical processing conditions, to keep processing and microstructure from adding variation. The best of those 22 had \(\Delta T = 3.15\) K. \(\Delta T\) was measured by differential scanning calorimetry (DSC) as the difference between the endothermic peak temperature on heating and the exothermic peak on cooling.

The model did not see compositions directly. Each alloy was described by six features drawn from metallurgical knowledge, including Pauling electronegativity, the valence electron number (VEN), the metallic radius and the Pettifor chemical scale. Each feature was computed as a composition-weighted average over the elements present. The authors chose them from prior work linking transformation temperatures to valence electron concentration and hysteresis to the atomic size of the dopants.

What happened#

The loop ran for nine iterations, synthesizing four alloys per iteration, for 36 new compositions in total. Fourteen of the 36 had \(\Delta T\) below 3.15 K, the best value in the starting set. The best was Ti\(_{50.0}\)Ni\(_{46.7}\)Cu\(_{0.8}\)Fe\(_{2.3}\)Pd\(_{0.2}\), found in the sixth iteration, with \(\Delta T = 1.84\) K by DSC. The paper reports this as 42% better than the best training alloy. Over 60 thermal cycles its transformation temperature shifted by about 0.02 K.

The campaign did not improve steadily. In the first iteration one of the four proposed alloys showed no martensitic transformation at all, while another gave the second-best \(\Delta T\), as the paper reports. The spread of measured \(\Delta T\) was large in iterations 1 and 2, narrowed through iterations 3 to 6, and widened again from iteration 7 on. Plotted against the average VEN, the loop concentrated near a VEN of about 6.95 in the middle iterations and then moved away from it. The authors state that after the sixth iteration their design “drifts away from the apparent global minimum”. They interpret the late spread as the selection rule leaving a region it had already sampled to look for other minima, and they note it could be misread as overfitting.

To check that the result was not luck, the authors compared the 36 designed alloys with the 22 original ones using a Mann-Whitney test and found the designed set significantly better (\(P < 0.001\)). They also computed that, if the designed alloys had been drawn from the same distribution as the originals, the chance that 14 or more of the lowest \(\Delta T\) values would land in the designed set is \(3.7 \times 10^{-4}\).

Why it happened#

The loop is the one in L15: fit a surrogate to everything measured so far, score every unmeasured candidate with an acquisition function, run the best-scoring experiments, add the results, and refit. Three choices adapted it to alloys.

A surrogate that is not a Gaussian process. The authors tried three regressors from scikit-learn: a Gaussian process model, and support vector regression (SVR) with a radial basis function kernel and with a linear kernel. They write that SVR predicted better on their data, but SVR gives only a point prediction. To get an uncertainty they used the bootstrap: resample the training alloys with replacement, refit, and use the spread. They predicted \(\Delta T\) for every alloy in the search space with 1,000 bootstrap samples and took the mean as \(\mu\) and the spread as \(\sigma\).

Expected improvement, written for a minimum. The paper’s EGO selector maximizes \(\sigma[\phi(z) + z\Phi(z)]\) with \(z = (\mu^* - \mu)/\sigma\), where \(\mu^*\) is the smallest \(\Delta T\) measured so far. They also tried a selector they call knowledge gradient (KG), which is the same formula with \(\mu^*\) replaced by the smaller of the best measured value and the best predicted value anywhere in the unexplored space. A greedy selector (Min) picked the lowest predicted \(\Delta T\), and a Random selector ignored the model.

A rule for picking the rule. With 22 data points there is no way to know in advance which regressor and selector pair will search well. The authors simulated the search inside their training set. They gave each pair a few randomly chosen alloys, let it pick the rest one at a time, and counted the picks needed to reach the best alloy, repeating this 2,000 times per pair and starting size. Some pairs did worse than random picking. The greedy GP pair was best when it started from 2 or 3 alloys and then fell behind. From 4 to 8 starting alloys, SVR with the radial kernel did best under both KG and EGO, with nearly identical results, and the authors chose SVR with KG.

Four alloys could be synthesized at a time, so the loop needed a batch. They built it with the kriging believer heuristic: take the top candidate, add its predicted \(\Delta T\) to the data as though it had been measured, refit, pick the next, and repeat four times.

Connections to the lectures#

The section on the Bayesian optimization loop in L15 defines the incumbent, the budget and the four-step loop, and all three appear here: the incumbent starts at 3.15 K, the budget spent was 36 syntheses, and the loop ran nine times. The L15 section on acquisition functions writes EI for a maximum as \((\mu - f^*)\Phi(z) + \sigma\phi(z)\). The paper’s form is the same expression for a minimum, with the sign of the gap flipped. Its \(\sigma z \Phi(z)\) term is the exploitation term and its \(\sigma\phi(z)\) term is the exploration term. The paper also describes the two limits the way L15’s section on exploration and exploitation does: as \(\sigma \to 0\) the rule picks the lowest predicted value, and as \(\sigma\) grows it picks the most uncertain candidates. The L15 subsection on Donald Jones and EGO is the paper’s own reference 7.

The two problems also differ. The reactor’s surrogate is a GP over one or two continuous inputs. Here the inputs are six composition-weighted features, the candidates are a finite list of 797,482 unmeasured alloys, and the surrogate predicts every one of them. The batch of four matches L15’s “Extensions engineers need”, where a lab with four reactors chooses four experiments at once; the kriging believer is one heuristic for that job. The 2,000-repeat comparison against random picking follows L15’s section on comparing optimizers over several seeds, and the pairs that lost to random show why a baseline belongs in that comparison. For the list in L15’s section “Where Bayesian optimization is used”, this is a materials entry, and a reminder that the loop runs on any surrogate that supplies a \(\mu\) and a \(\sigma\).

The uncertainty came from the section on bootstrap and ensembles in L14, and that section’s caveats apply directly. A bootstrap spread estimates how much the fitted function would change with a new data set of the same size. It is a confidence interval, so it leaves out measurement noise, and it can be small where the model is biased, which L14 warns happens far outside the domain of the data. With 22 training alloys and nearly 800,000 candidates, most of the search space is outside that domain. L14’s limitations section adds that a design loop creates points unlike its training data on purpose. The main text of the paper does not report a calibration check of the bootstrap \(\sigma\), so how well the spread tracked real error is not known from it.

L9 supplies two more pieces. SVR is not one of L9’s four model families, but L9’s section on those families gives the reason for comparing several: no model wins on every problem. The authors cite a no-free-lunch theorem for the same reason (Wolpert and Macready’s version for optimization, not the supervised-learning version in L9). The six descriptors are feature engineering in the sense of L9’s section on linear regression, with knowledge of the system put into the features.

What a practitioner should take from this

  • Any regressor can drive the loop if you can attach a \(\sigma\) to it. A bootstrap over a model that predicts well on your data is a legitimate choice, but it measures variance only, so check it against held-out measurements before you trust it far from the data.

  • Choose the surrogate and acquisition function by simulating the search on data you already have, with many random restarts and a random-picking baseline. Expect some combinations to lose to random.

  • If you can run several experiments at once, plan the batch explicitly (kriging believer, or a batch acquisition such as qLogEI) rather than taking the top four EI scores, which can sit next to each other.

  • Read a widening spread late in a campaign against your acquisition function before calling it overfitting. An EI-based rule is built to leave a region it has already exploited.

  • Decide every property you need before the campaign. This loop optimized \(\Delta T\) alone, and the paper reports that its alloys came out with relatively high transformation temperatures, which were not designed for.

  • Report the incumbent at each iteration, the number of syntheses, and a test against the starting data, as this paper does, so a reader can tell search from luck.

Sources#