Black-Box Optimization of Model Superconducting Materials¶

To illustrate the effective use of black-box optimization, this sample code describes the optimization for superconducting materials composed of pseudo-materials as an example problem. Although the present sample code performs material searches based on nonlinear algebraic models, you can perform black-box optimization with the same steps based on high-precision simulations or experimental measurement results instead of model algebraic expressions. Even in such cases, you can use this example almost as is.

For a basic introduction to black-box optimization and FMQA, see "Black-Box Optimization with Quantum Annealing and Ising Machines".

Also, you can find example programs of FMQA applications in various fields such as material search, fluid engineering, chemical plant and urban transportation.

Problem setting¶

Search scenario for superconducting materials¶

Superconductivity technology is expected to be utilized in the fields of transportation, such as maglev trains, metrology, and energy. Various superconducting materials are currently being developed to realize superconductivity.

The temperature at which superconductivity is achieved (critical temperature) is generally around the absolute temperature of 0 K (Kelvin) for currently confirmed superconducting materials. Because of this, superconductivity requires costly cooling to be exploited, and its application in the real world is currently limited. Therefore, the search for high-temperature superconductors is a pressing issue.

Typically, the search for materials that realize superconductivity involves a trial-and-error process of selecting and synthesizing several materials, repeatedly evaluating the critical temperature of the synthesized materials by measurement, and identifying the material to be synthesized that achieves a higher critical temperature. This process of synthesis and critical temperature evaluation is considered time-consuming. For this search, a black box optimization method is used to find a combination of materials close to the optimal solution with a relatively small number of evaluations.

In this example, the search for superconducting materials consisting of pseudo materials is treated as an example to illustrate the material search by a black-box optimization method (FMQA), and a critical temperature model is used to evaluate the critical temperature. Note that the critical temperature models presented below and the combinations of materials obtained are not necessarily physically accurate, and thus the example serves for illustration purposes only.

Solver client configuration¶

First, we will configure the solver client of the solver we will use during the optimization cycles. In this example program, we will use Amplify AE.

Definition of black-box function¶

This example code selects a combination of several materials from $D$ types of materials and performs an optimization to maximize the critical temperature of the superconducting material produced by their synthesis.

In general, the critical temperature can be evaluated by experimental measurement, which requires a relatively large cost (time and money) each time the evaluation is performed. In this example code, instead of measuring the critical temperature, the following critical temperature model supercon_temperature() is used for evaluation. However, this function is only a substitute for experimental measurement, and its contents and parameters are treated as unknown, and the number of calls to supercon_temperature() is also treated as limited.

The following make_blackbox_func function creates and returns a blackbox function blackbox, which is also the objective function in the present FMQA. The blackbox function blackbox executes supercon_temperature() and returns the negative value of the obtained critical temperature.

In this example, optimization will progress to minimize the negative value of the obtained critical temperature.

In the following, the black-box function blackbox(x) for the critical temperature defined above is used to evaluate the critical temperature of superconducting materials synthesized from a random selection of materials. Here, num_materials is the number of materials to be selected, and the input x is a vector of size num_materials consisting of 0 or 1.

For example, in the case of selecting the first and last of five materials to be combined, the input vector would be x = [1, 0, 0, 0, 0, 1]. In this case, there are $2^5-1=31$ possible choices (combinations).

For num_materials = 100, the number of combinations is approximately $10^{30}$, and the full-search method is considered difficult.

FMQA program implementation¶

Model training by machine learning¶

This section implements the part of FMQA that learns the optimal parameters (weights and bias) of the model by machine learning. First, the TorchFM class representing the model by the Factorization Machine is defined using PyTorch.

The following equation represents the Factorization Machine.

$$ \begin{aligned} f(\boldsymbol{x} | \boldsymbol{w}, \boldsymbol{v}) &= \underset{\color{red}{\mathtt{out\_linear}}}{\underline{ w_0 + \sum_{i=1}^d w_i x_i} } + \underset{\color{red}{\mathtt{out\_quadratic}}}{\underline{\frac{1}{2} \left[\underset{\color{red}{\mathtt{out\_1}}}{\underline{ \sum_{f=1}^k\left(\sum_{i=1}^d v_{i f} x_i\right)^2 }} - \underset{\color{red}{\mathtt{out\_2}}}{\underline{ \sum_{f=1}^k\sum_{i=1}^d v_{i f}^2 x_i^2 }} \right] }} \end{aligned} $$

The input $x$ of this model is a vector of the length $d$ as the input to the black-box function, with the following three parameters.

• $v$: 2-dimensional array of $d\times k$.
• $w$: 1D vector of length $d$.
• $w_0$: scalar

The only hyperparameter is $k$, which is given as a positive integer less than or equal to $d$.

The TorchFM class defined below inherits from torch.nn.Module and is constructed from an input vector $x$ of size $d$ and a hyperparameter $k$. The hyperparameter $k$ controls the number of parameters in the model; the larger the hyperparameter, the more parameters are, the more accurate the model becomes, but also the more prone to over-fitting.

The TorchFM class has attributes $v$, $w$, and $w_0$ of the model parameters and updates these parameters as the training proceeds. According to the above formula, the forward method also outputs an estimate of $y$ from the input $x$. Since the parameters $v$, $w$, and $w_0$ are needed to construct a QUBO model later, we also define a function get_parameters to output these parameters.

Next, we define a function train() to perform machine learning of the model based on the TorchFM class. The input is the training data $x, y$ and an instance of the TorchFM model. By calling the train() function, the TorchFM parameters are trained and optimized.

In general machine learning, we split the data into training and validation data, optimize the parameters using the training data, and validate the model during training using the validation data. The model is validated at each epoch, and the parameters at the epoch with the highest prediction accuracy for the validation data are saved and used as the model after learning.

Solving the model with Amplify¶

Next, we implement the anneal function to minimize the inferred machine learning model, corresponding to the lower left part of the FMQA cycle diagram (reproduced below), where the input is the model TorchFM class after training, and the output is a vector $x$ that minimizes the model.

Solve the following optimization problem for the class TorchFM model learned earlier to find the input $x$ such that the inferred model is minimized.

$$ \underset{x}{\mathrm{argmin}} \quad \underset{\color{red}{\mathtt{out\_linear}}}{\underline{ w_0 + \sum_{i=1}^d w_i x_i} } + \underset{\color{red}{\mathtt{out\_quadratic}}}{\underline{\frac{1}{2} \left[\underset{\color{red}{\mathtt{out\_1}}}{\underline{ \sum_{f=1}^k\left(\sum_{i=1}^d v_{i f} x_i\right)^2 }} - \underset{\color{red}{\mathtt{out\_2}}}{\underline{ \sum_{f=1}^k\sum_{i=1}^d v_{i f}^2 x_i^2 }} \right] }} $$

The decision variable in this optimization problem is $x$. It is a one-dimensional binary variable vector of length $d$, just like the input vector to the black-box function. Also, $v$, $w$, and $w_0$, parameters in the learning phase, are constants here.

We define the anneal function that performs the optimization using Amplify AE for the given model as follows. The anneal function uses the VariableGenerator to create a 1D binary vector x of length $d$, and then uses the binary variable array $x$ and $v$, $w$, and $w_0$ obtained from the TorchFM class to create a Factorization Machine The objective function to be optimized is created according to the formula of Factorization Machine.

After building the optimization model, we will perform minimization of the objective function using the created QUBO model and the solver client FixstarsClient defined earlier.

Now that we have defined the train function for machine learning and the anneal function for optimization, which are the core of FMQA, we can execute FMQA using these functions.

The black-box function (corresponding to experiment or simulation) to be used for th epresent black-box optimization is the blackbox_func() already defined above. This function takes a NumPy one-dimensional binary vector of length $d = 100$ consisting of $0$ or $1$ and returns the negative value of the critical temperature.

Creating initial training data¶

Next, the black-box function $y = f(\boldsymbol{x})$ is evaluated on the input vector $\boldsymbol{x}$ to create $N_0$ initial training data. The black-box function corresponds to the results of a simulation or experiment, so you can create these data based on previous experiments or simulations.

We define the init_training_data function to create the initial training data using a random $N_0$ input vector $x$ as follows. This function takes the black-box function and the number of initial training data $N_0$ and returns $N_0$ input vectors $\boldsymbol{x}$ and corresponding output $y$ as initial training data.

Now, we have $N_0$-pair of initial training data (10 pairs of the input vector of size 100 and the output value).

Running an FMQA cycle¶

Using the $N_0$ pairs of initial training data created above, we will run the FMQA cycles, by performing the following operations per FMQA cycle.

1. Train the model
   • Construct a model of class TorchFM and train it by calling the train function on the model with initial training data x = x_init, y = y_init.
2. Minimize the model
   • Minimize the model by passing a trained model of class TorchFM to the anneal function to get $\hat{x}$ that minimizes the model.
   • If $\hat{x}$ is already included in the training data x, change part of $\hat{x}$ to avoid duplicate training data.
3. Evaluation of black-box function
   • Input $\hat{x}$ to the black-box function and get output $\hat{y}$.
   • Add $\hat{x}$ and $\hat{y}$ to the traiing data x and y, respectively.

An example implementation of the above is as follows: Since the execution takes approximately 10 minutes of computation time, an example output is shown in Execution example.

After running the above cell, x and y have input-output pairs of black-box function evaluations. The number of pairs is $N_0 + N = 50$.

The following gives the input that gives the minimum value of the black box function evaluation and its value.

Plotting the evolution of the evaluated values¶

Below are $N_0$ initial training data and the evolution of the optimized evaluation values of the black-box function over $N$ FMQA cycles. The initial training data is shown in blue, and the evaluation values obtained by the FMQA cycles are shown in red.

Execution example¶

The output of a typical plot obtained by running this sample code is shown below. The optimization uses a QUBO solver, so the results will vary from run to run. However, you can see that the evaluated value of the black-box function decreases with the number of FMQA cycles.