BOIS: Bayesian Optimization of Interconnected Systems

Bayesian optimization (BO) has proven to be an effective paradigm for the global optimization of expensive-to-sample systems. One of the main advantages of BO is its use of Gaussian processes (GPs) to characterize model uncertainty which can be leveraged to guide the learning and search process. However, BO typically treats systems as black-boxes and this limits the ability to exploit structural knowledge (e.g., physics and sparse interconnections). Composite functions of the form $f(x, y(x))$, wherein GP modeling is shifted from the performance function $f$ to an intermediate function $y$, offer an avenue for exploiting structural knowledge. However, the use of composite functions in a BO framework is complicated by the need to generate a probability density for $f$ from the Gaussian density of $y$ calculated by the GP (e.g., when $f$ is nonlinear it is not possible to obtain a closed-form expression). Previous work has handled this issue using sampling techniques; these are easy to implement and flexible but are computationally intensive. In this work, we introduce a new paradigm which allows for the efficient use of composite functions in BO; this uses adaptive linearizations of $f$ to obtain closed-form expressions for the statistical moments of the composite function. We show that this simple approach (which we call BOIS) enables the exploitation of structural knowledge, such as that arising in interconnected systems as well as systems that embed multiple GP models and combinations of physics and GP models. Using a chemical process optimization case study, we benchmark the effectiveness of BOIS against standard BO and sampling approaches. Our results indicate that BOIS achieves performance gains and accurately captures the statistics of composite functions.

Uncertainty Quantification for Molecular Property Predictions with Graph Neural Architecture Search

Graph Neural Networks (GNNs) have emerged as a prominent class of data-driven methods for molecular property prediction. However, a key limitation of typical GNN models is their inability to quantify uncertainties in the predictions. This capability is crucial for ensuring the trustworthy use and deployment of models in downstream tasks. To that end, we introduce AutoGNNUQ, an automated uncertainty quantification (UQ) approach for molecular property prediction. AutoGNNUQ leverages architecture search to generate an ensemble of high-performing GNNs, enabling the estimation of predictive uncertainties. Our approach employs variance decomposition to separate data (aleatoric) and model (epistemic) uncertainties, providing valuable insights for reducing them. In our computational experiments, we demonstrate that AutoGNNUQ outperforms existing UQ methods in terms of both prediction accuracy and UQ performance on multiple benchmark datasets. Additionally, we utilize t-SNE visualization to explore correlations between molecular features and uncertainty, offering insight for dataset improvement. AutoGNNUQ has broad applicability in domains such as drug discovery and materials science, where accurate uncertainty quantification is crucial for decision-making.

A Graph-Based Modeling Framework for Tracing Hydrological Pollutant Transport in Surface Waters

Anthropogenic pollution of hydrological systems affects diverse communities and ecosystems around the world. Data analytics and modeling tools play a key role in fighting this challenge, as they can help identify key sources as well as trace transport and quantify impact within complex hydrological systems. Several tools exist for simulating and tracing pollutant transport throughout surface waters using detailed physical models; these tools are powerful, but can be computationally intensive, require significant amounts of data to be developed, and require expert knowledge for their use (ultimately limiting application scope). In this work, we present a graph modeling framework -- which we call ${\tt HydroGraphs}$ -- for understanding pollutant transport and fate across waterbodies, rivers, and watersheds. This framework uses a simplified representation of hydrological systems that can be constructed based purely on open-source data (National Hydrography Dataset and Watershed Boundary Dataset). The graph representation provides an flexible intuitive approach for capturing connectivity and for identifying upstream pollutant sources and for tracing downstream impacts within small and large hydrological systems. Moreover, the graph representation can facilitate the use of advanced algorithms and tools of graph theory, topology, optimization, and machine learning to aid data analytics and decision-making. We demonstrate the capabilities of our framework by using case studies in the State of Wisconsin; here, we aim to identify upstream nutrient pollutant sources that arise from agricultural practices and trace downstream impacts to waterbodies, rivers, and streams. Our tool ultimately seeks to help stakeholders design effective pollution prevention/mitigation practices and evaluate how surface waters respond to such practices.

Convolutional Neural Networks: Basic Concepts and Applications in Manufacturing

We discuss basic concepts of convolutional neural networks (CNNs) and outline uses in manufacturing. We begin by discussing how different types of data objects commonly encountered in manufacturing (e.g., time series, images, micrographs, videos, spectra, molecular structures) can be represented in a flexible manner using tensors and graphs. We then discuss how CNNs use convolution operations to extract informative features (e.g., geometric patterns and textures) from the such representations to predict emergent properties and phenomena and/or to identify anomalies. We also discuss how CNNs can exploit color as a key source of information, which enables the use of modern computer vision hardware (e.g., infrared, thermal, and hyperspectral cameras). We illustrate the concepts using diverse case studies arising in spectral analysis, molecule design, sensor design, image-based control, and multivariate process monitoring.

New Paradigms for Exploiting Parallel Experiments in Bayesian Optimization

Bayesian optimization (BO) is one of the most effective methods for closed-loop experimental design and black-box optimization. However, a key limitation of BO is that it is an inherently sequential algorithm (one experiment is proposed per round) and thus cannot directly exploit high-throughput (parallel) experiments. Diverse modifications to the BO framework have been proposed in the literature to enable exploitation of parallel experiments but such approaches are limited in the degree of parallelization that they can achieve and can lead to redundant experiments (thus wasting resources and potentially compromising performance). In this work, we present new parallel BO paradigms that exploit the structure of the system to partition the design space. Specifically, we propose an approach that partitions the design space by following the level sets of the performance function and an approach that exploits partially-separable structures of the performance function found. We conduct extensive numerical experiments using a reactor case study to benchmark the effectiveness of these approaches against a variety of state-of-the-art parallel algorithms reported in the literature. Our computational results show that our approaches significantly reduce the required search time and increase the probability of finding a global (rather than local) solution.

SAFE-OCC: A Novelty Detection Framework for Convolutional Neural Network Sensors and its Application in Process Control

We present a novelty detection framework for Convolutional Neural Network (CNN) sensors that we call Sensor-Activated Feature Extraction One-Class Classification (SAFE-OCC). We show that this framework enables the safe use of computer vision sensors in process control architectures. Emergent control applications use CNN models to map visual data to a state signal that can be interpreted by the controller. Incorporating such sensors introduces a significant system operation vulnerability because CNN sensors can exhibit high prediction errors when exposed to novel (abnormal) visual data. Unfortunately, identifying such novelties in real-time is nontrivial. To address this issue, the SAFE-OCC framework leverages the convolutional blocks of the CNN to create an effective feature space to conduct novelty detection using a desired one-class classification technique. This approach engenders a feature space that directly corresponds to that used by the CNN sensor and avoids the need to derive an independent latent space. We demonstrate the effectiveness of SAFE-OCC via simulated control environments.

Convolutional Neural Nets: Foundations, Computations, and New Applications

We review mathematical foundations of convolutional neural nets (CNNs) with the goals of: i) highlighting connections with techniques from statistics, signal processing, linear algebra, differential equations, and optimization, ii) demystifying underlying computations, and iii) identifying new types of applications. CNNs are powerful machine learning models that highlight features from grid data to make predictions (regression and classification). The grid data object can be represented as vectors (in 1D), matrices (in 2D), or tensors (in 3D or higher dimensions) and can incorporate multiple channels (thus providing high flexibility in the input data representation). For example, an image can be represented as a 2D grid data object that contains red, green, and blue (RBG) channels (each channel is a 2D matrix). Similarly, a video can be represented as a 3D grid data object (two spatial dimensions plus time) with RGB channels (each channel is a 3D tensor). CNNs highlight features from the grid data by performing convolution operations with different types of operators. The operators highlight different types of features (e.g., patterns, gradients, geometrical features) and are learned by using optimization techniques. In other words, CNNs seek to identify optimal operators that best map the input data to the output data. A common misconception is that CNNs are only capable of processing image or video data but their application scope is much wider; specifically, datasets encountered in diverse applications can be expressed as grid data. Here, we show how to apply CNNs to new types of applications such as optimal control, flow cytometry, multivariate process monitoring, and molecular simulations.

Dynamic penalty function approach for constraints handling in reinforcement learning

Reinforcement learning (RL) is attracting attentions as an effective way to solve sequential optimization problems involving high dimensional state/action space and stochastic uncertainties. Many of such problems involve constraints expressed by inequalities. This study focuses on using RL to solve such constrained optimal control problems. Most of RL application studies have considered inequality constraints as soft constraints by adding penalty terms for violating the constraints to the reward function. However, while training neural networks to represent the value (or Q) function, a key step in RL, one can run into computational issues caused by the sharp change in the function value at the constraint boundary due to the large penalty imposed. This difficulty during training can lead to convergence problems and ultimately poor closed-loop performance. To address this problem, this study suggests the use of a dynamic penalty function which gradually and systematically increases the penalty factor during training as the iteration episodes proceed. First, we examined the ability of a neural network to represent an artificial value function when uniform, linear, or dynamic penalty functions are added to prevent constraint violation. The agent trained by a Deep Q Network (DQN) algorithm with the dynamic penalty function approach was compared with agents with other constant penalty functions in a simple vehicle control problem. Results show that the dynamic penalty approach can improve the neural network's approximation accuracy and that brings faster convergence to a solution closer to the optimal solution.

Overlapping Schwarz Decomposition for Nonlinear Optimal Control

We present an overlapping Schwarz decomposition algorithm for solving nonlinear optimal control problems (OCPs). Our approach decomposes the time domain into a set of overlapping subdomains and solves subproblems defined over such subdomains in parallel. Convergence is attained by updating primal-dual information at the boundaries of the overlapping regions. We show that the algorithm exhibits local convergence and that the convergence rate improves exponentially with the size of the overlap. Our convergence results rely on a sensitivity result for OCPs that we call "asymptotic decay of sensitivity." Intuitively, this result states that impact of parametric perturbations at the boundaries of the time domain (initial and final time) decays exponentially as one moves away from the perturbation points. We show that this condition holds for nonlinear OCPs under a uniform second-order sufficient condition, a controllability condition, and a uniform boundedness condition. The approach is demonstrated by using a highly nonlinear quadrotor motion planning problem.

Unifying Theorems for Subspace Identification and Dynamic Mode Decomposition

This paper presents unifying results for subspace identification (SID) and dynamic mode decomposition (DMD) for autonomous dynamical systems. We observe that SID seeks to solve an optimization problem to estimate an extended observability matrix and a state sequence that minimizes the prediction error for the state-space model. Moreover, we observe that DMD seeks to solve a rank-constrained matrix regression problem that minimizes the prediction error of an extended autoregressive model. We prove that existence conditions for perfect (error-free) state-space and low-rank extended autoregressive models are equivalent and that the SID and DMD optimization problems are equivalent. We exploit these results to propose a SID-DMD algorithm that delivers a provably optimal model and that is easy to implement. We demonstrate our developments using a case study that aims to build dynamical models directly from video data.