Daniel
Frey
^{a},
Ju Hee
Shin
^{a},
Christopher
Musco
^{b} and
Miguel A.
Modestino
*^{a}
^{a}Department of Chemical and Biomolecular Engineering, Tandon School of Engineering, New York University, 6 Metrotech Center, Brooklyn, NY, USA. E-mail: modestino@nyu.edu; Tel: +1 646 997 3594
^{b}Computer Science and Engineering, Tandon School of Engineering, New York University, 370 Jay Street, Brooklyn, NY, USA

Received
6th January 2022
, Accepted 24th February 2022

First published on 4th March 2022

Current methods of finding optimal experimental conditions, Edisonian systematic searches, often inefficiently evaluate suboptimal design points and require fine resolution to identify near optimal conditions. For expensive experimental campaigns or those with large design spaces, the shortcomings of the status quo approaches are more significant. Here, we extend Bayesian optimization (BO) and introduce a chemically-informed data-driven optimization (ChIDDO) approach. This approach uses inexpensive and low-fidelity information obtained from physical models of chemical processes and subsequently combines it with expensive and high-fidelity experimental data to optimize a common objective function. Using common optimization benchmark objective functions, we describe scenarios in which the ChIDDO algorithm outperforms the traditional BO approach, and then implement the algorithm on a simulated electrochemical engineering optimization problem.

Bayesian optimization (BO) has been widely implemented in different fields of research to accelerate experimental optimization.^{3–9} BO methods use a surrogate model (SM) that describes an objective function and its probability distribution in the design space to guide the optimization campaign. Each time new experimental data is obtained, the SM is updated to increase its accuracy. In this way, Bayesian statistics and reasoning can be used to select the most informative sequence of experiments and accelerate optimization campaigns.^{10} In recent years, BO has been implemented for various applications in the chemical sciences including materials discovery and prediction of their properties,^{11–21} design of reactors and chemical processes,^{22–33} and the optimization of energy storage materials and devices.^{34–38} Data-driven optimization methods such as BO learn and evolve with new experimental data, but they lack a priori knowledge of the physical laws that dictate the behavior of the chemical system under study. This can result in the need for large experimental campaigns to accurately model and find the optimal combination of parameters for a given objective function. On the other hand, physical models (e.g., density functional theory, molecular dynamics, continuum models, etc.) could be used to identify optima without the need to perform experimental searches, but they often lack the accuracy to effectively capture the complexity of real systems or require inaccessibly-large computational power. Given the advantages and shortcomings of both optimization approaches, there is an opportunity to leverage a priori chemical knowledge in data-driven optimization to reduce the data needs and allow for faster identification of optima.

Herein, we introduce a chemically-informed data-driven optimization (ChIDDO) approach, which is a type of multi-information source optimization (MISO), where inexpensive and low-fidelity information obtained from physical models of chemical processes are combined with high-fidelity experimental data to optimize a common objective function. In this study, we leverage simulated data to develop a ChIDDO approach that can be implemented broadly in experimental campaigns. While MISO algorithms have been previously implemented to improve BO in computational problems,^{39–41} the implementation of ChIDDO can extend these advantages to chemical experimentation. In addition, we introduce a new acquisition function, modified ranked batch (MRB) that could improve the selection of a batch of experiments.^{42}

An acquisition function is used to select the next design condition(s) to evaluate, x^{next}, based on how informative the design conditions will be in the goal of optimizing the cost function. Here, we can choose to select a single design condition or a batch of conditions. In the chemical sciences, it is often convenient to run multiple experiments in parallel based on equipment capabilities, so we chose to focus on selecting batches of design conditions. Many different acquisition functions for BO have been developed, and three of the most common are expected improvement (EI),^{43} probability of improvement (PI),^{44} and upper confidence bound (UCB).^{45} In addition to these, we have developed a modified ranked-batch (MRB) mode sampling function inspired by the work of Cardoso et al.^{42} The equations for each of the acquisition functions are provided in the ESI.† An acquisition function uses the current information, X^{exp} and y^{exp}, and the SM predictions to calculate how informative a possible design condition is expected to be based on the criteria for the respective acquisition function. To determine the most informative design point to sample next, a maximization method was used to find a local maximum of the acquisition function score. This process was repeated 25 times at different initiation points to get closer to the global maximum solution. The design point with the maximum score was subsequently added to x^{next}. For this study a minimization method was used and the negative of the acquisition function score was minimized. The minimization method was the L-BFGS-B method from the scipy.optimize.minimize package. Depending on the batch size used in the optimization campaign, n_{b}, multiple design conditions can be added to x^{next} by repeating this acquisition function maximization step. After x^{next} is selected, the experimental objective function value(s) are determined to obtain y^{next}. Subsequently, x^{next} and y^{next} are appended to x^{exp} and y^{exp}.

The EI, PI, and UCB algorithms were run based on their implementation in the modAL active learning framework,^{46} which is described in the ESI.† The general framework for the BO algorithms presented was also based on the modAL framework. The MRB acquisition function calculated a score consisting of three normalized parameters: a distance score, Δ, an uncertainty score, Γ, and the objective function prediction, Ω. The distance score was calculated as:

(1) |

Score = βΔ + βΓ + Ω | (2) |

To initiate the algorithm, N_{init} evenly distributed random points were chosen as the initial set of experimental conditions. Our results show robust performance when a random initialization approach is implemented, but other methods that ensure good spatial coverage over the design space and incorporate a degree of randomness could be implemented. The random initialization approach was done by choosing random experiments to perform without considering the positions of the other initial experiments. In other words, there was no space-filling model for this initialization approach. For the BO algorithm without the use of a physics model (referred to as BO from this point on), only these initial points, x^{exp}_{init}, were fit by the GPR to generate the SM. After each batch of BO, (x^{exp}, y^{exp}) increases in size by the batch size, n_{b}. For the ChIDDO algorithm, before (x^{exp}, y^{exp}) are passed to the GPR, a certain number of design points from the a priori physics model (x^{phys}, y^{phys}) are appended to (x^{exp}, y^{exp}). The size of (x^{phys}, y^{phys}) decreases as the number of experiments that are run increases. For example, if it was decided that a total of 50 experiments would be run before stopping the ChIDDO optimization campaign, N_{total}, and it was chosen to start with 10 experimental points, the ChIDDO algorithm would add (N_{total} − size(x^{exp})) data points (40 in this case) calculated from the physics model. These added points were uniformly distributed random design points between the upper and lower bounds. This method allowed for the incorporation of knowledge of the chemical system under study to help guide the initial choice of experiments when less experimental data is available, and progressively increases the amount of experimental data used to generate an SM as more empirical evidence becomes available. A general algorithm flowchart is shown in Fig. 1 and an example of the decision process in action is shown in Fig. 2.

(3) |

Depending on the specific objective function, we studied 2-, 3-, 4- and 6-dimensional spaces. Unless otherwise specified, the experimental objective function values, y^{exp}, were exactly equal to the objective function calculation, given the set of parameter values.

Under conditions when noise was added, the objective function values were calculated as:

y^{noise}_{i} = y^{exp}_{i} + [(y_{max} − y_{min})(2rand(0,1) − 1)]η | (4) |

y^{mixed} = ry_{1} + (1 − r)y_{2} | (5) |

Fig. 3 Example of the addition of two dissimilar objective functions. The Rosenbrock function and sphere function are shown using their respective base case parameters. |

A + B + e^{−} → C | (6) |

2A + B + e^{−} → D | (7) |

3A + B + e^{−} → E | (8) |

2B + e^{−} → F | (9) |

The chosen reaction resembles the electrohydrodimerization of acrylonitrile to adiponitrile, the largest organic electrosynthetic process practices in industry.^{1,47,48}

The rates of these reactions were modelled by Butler–Volmer kinetics in the form:

(10) |

The reactions are simulated in a 1-D domain, representing the diffusion boundary layer, on one end bounded by the bulk electrolyte solution and the other end the electrode surface. The Nernst–Planck equation was used to model the concentration change of each species using diffusion, migration, and generation terms:

(11) |

The Faradaic efficiency (FE) of product D was the value to be optimized. FE is a metric that measures how much of the current participates in the desired reaction. In this case, FE is calculated by dividing the amount of D produced by the total of all produced species (including D). In this system, the concentrations of the reactants could have a large effect on the FE. Therefore, the optimization variables in the 2D design space for this reaction were the bulk concentrations of reactants A and B. Due to the reaction rates and reaction orders of the different reactions, an optimal set of reactant concentrations could be located in the design space.

Fig. 4
d
_{
y
}
versus number of experiments, N, comparing BO and ChIDDO with the Edisonian random and grid search. (A) 2D sphere, (B) 3D sphere, (C) 4D sphere. For each curve, 20 separate searches, S, were performed, and the average of the results are the lines shown. The shadow around each of the lines represents the standard deviation. For each of the BO/ChIDDO experiments, the MRB acquisition function was used. The objective function parameter information is provided in the ESI.† |

In our framework, we consider experimental sets, S, which consist of N^{exp} number of experiments with conditions x^{exp} resulting in output performance, y^{exp}. The purpose of the BO algorithm is to maximize y^{exp} in the fewest number of experiments. The output of the algorithm generates a set of (x^{exp}, y^{exp}) results that can be plotted and compared to Edisonian experimental sets that follow either a grid or a random search approach. We evaluate two performance metrics: the normalized deviation from the optimum value, d_{y}, and the minimum distance from the optimum, d_{x}, identified by each set of experiments. These two quantities are calculated as,

(12) |

(13) |

In the following studies, the different algorithms (BO and ChIDDO) are run on 20 different sphere functions which serve as simulated experimental objective functions. Since the experimental objective functions are different, there was some variance in the results between the 20 runs. Therefore, the graphs shown in the following figures show the average of the 20 runs as a solid line, and a shadow around the solid line representing the standard deviation of the 20 runs. In Fig. 4, d_{y} is plotted against the number of experiments, N, comparing the Edisonian methods with BO and ChIDDO. The plots for d_{x} can be found in the ESI.†Fig. 4A shows how the different search algorithms compare using the 2D sphere objective function. Even for this simple, parabolic function, the systematic grid search and random search underperform comparatively to BO or ChIDDO. d_{y} values after 30 experiments, d_{y30}, were 0.008 and 0.023 for the grid and random search algorithms, respectively. In comparison, d_{y30} for BO and ChIDDO were both on the order of 10^{−3}. As the design space moves to higher dimensions, Fig. 4B and C show that the differences between the algorithms increase with dimension size. For the 3D sphere objective function the enhancements are more drastic, with d_{y30} being two orders of magnitude smaller for BO compared to the Edisonian algorithms. Because of the larger design space to sample, the grid and random search methods are not capable of searching a fine enough space to find values close to the optimal. It is of interest that the d_{y30} for BO and ChIDDO were very similar, possibly because the sphere objective function has a well-defined optimum and is therefore easy to identify. As the number of dimensions increases, ChIDDO tends to find near optimal values with fewer experiments than BO. This enhancement is likely because ChIDDO relies on the physics model initially to help more rapidly locate optimal conditions.

Fig. 5C and D show the comparison on the 4D Hartmann function using BO and ChIDDO, respectively. Interestingly, all the acquisition functions perform similarly with d_{y30} values on the order of 10^{−3} or lower and they all reach low values very quickly. The comparison on the 6D Hartmann function is shown in Fig. 5E and F. When using BO, all the acquisition functions appear to perform similarly with d_{y30} values of 0.218 (MRB), 0.134 (EI), 0.195 (PI), and 0.205 (UCB). It is important to note that the standard deviations for the 6D graphs are much larger than for the smaller dimensions. This indicates that the different random starting conditions affected the d_{y} values more for the 6D space compared with the 3D and 4D spaces, due to the larger complexity of the optimization process with increased dimensionality. Fig. 6F shows the comparison using ChIDDO. The d_{y30} values for MRB, EI, PI, and UCB were 0.079, 0.021, 0.027, and 0.149, respectively. In addition, the standard deviation of d_{y} is much smaller for ChIDDO than for BO, indicating a more consistent optimization.

When comparing the performance of BO to ChIDDO, it appears that the ChIDDO algorithm performs similarly or better for all of the objective functions. These results show that the ChIDDO algorithm does improve the performance initially, since the physics model information has a larger impact when fewer experiments are available.

Fig. 6 compares d_{y} for different levels of noise using the BO and ChIDDO algorithms with the MRB acquisition function. For the case of the 3D Hartmann using BO (Fig. 6A), the d_{y} for the highest noise level studied (i.e. η = 0.1) appears to be slightly higher than the other noise values until about the 37th experiment when d_{y} approaches the same value for all noise levels. For the 3D Hartmann function using ChIDDO in Fig. 6B, the observations are similar to that of BO as the noise had only a small impact on the optimization. Interestingly, the d_{y} values for the experiments with noise are not substantially different to the experiments without noise, demonstrating the robustness of BO and ChIDDO.

This behavior is also observed for the case of 4D Hartmann function in Fig. 6C and D. When using BO, the d_{y} for η = 0.1 remains higher than for the other noise levels until approximately the 32nd experiment when the d_{y} values start to converge for other noise levels. In the case of the 4D Hartmann function using ChIDDO, the d_{y} values for each noise level are similar after the 25th experiment. Prior to this, the d_{y} values for η = 0.1 are higher than that of the other noise levels. Contrary to the 3D Hartmann function, the experiments with no noise for both ChIDDO and BO have lower d_{y} values than the experiments with noise.

Fig. 6E and F show the noise comparisons for the 6D Hartmann function. When using BO, all noise levels present similar values for d_{y} until experiment 30th, and a slightly higher values for η = 0.1 beyond that point. These results indicate that the BO algorithm may be more resistant to noise effects in low-dimensionality design spaces and that overall noise effects are weak within the levels studied. Fig. 6F shows that the ChIDDO algorithm performs much better overall for the 6D Hartmann function compared to BO. When ChIDDO is implemented on 3, 4 and 6D Hartmann functions, our observations suggest that noise has only a small impact on the optimization but the values of d_{y} are lower than those found with BO for a given number of experiments.

The electrochemical model used in this study has 4 different parallel reactions (eqn (6)–(9)). It is common when simulating a complex reaction network that not all the intermediates or products are known. To test the robustness of the ChIDDO algorithm to incomplete physics models, eqn (8) and/or (9) was removed from the set of physics model reactions. When the full model was used as the physics model, a continuous improvement can be seen as more experiments are incorporated, as seen in Fig. 9. However, when one or two reactions are not included in the physics model, the algorithm is not able to improve the optimal value after the first few experiments. This could be the case if the simplified physics model does not agree with the values of the true objective function, leading to experimental selections that are far from the optimal values. After observing the model data from the simplified models, the objective values for the design space have different shapes and magnitudes than the experimental objective function. Examples of the simplified physics model data are shown in the ESI.† After a large number of experiments, the GPR prediction starts to become dominated by the experimental results and the exploration rate decreases, ultimately prompting the algorithm to select suboptimal experiments in close proximity to regions with low d_{y} values found during the early stage of the optimization. This indicates that it is important to have high accuracy in the physics model, or to extend the exploration phase of the algorithm if the information used in the physics model has large uncertainty.

Fig. 9
d
_{
y
}
versus number of experiments, N, for different electrochemical physics model information. “Full” indicates the model is predicting the same information as the objective function. “No E”, “No F”, and “No EF” indicate the removal of eqn (8) and/or (9) from the physics model information, resulting in a less informative model. (A) 2D electrochemical model. (B) 3D electrochemical model. (C) 4D electrochemical model. For each curve, 25 separate searches, S, were performed, and the average of the results are the lines shown. The shadow around each of the lines represents the standard deviation. For all of these graphs, ChIDDO was used as the AL algorithm and MRB was used as the acquisition function. |

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## Footnote |

† Electronic supplementary information (ESI) available: Acquisition function descriptions, objective function descriptions, and supplementary results. See DOI: 10.1039/d2re00005a |

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