Machine learning algorithms are increasingly being applied to fault detection and diagnosis (FDD) in chemical processes. However, existing data-driven FDD platforms often lack interpretability for process operators and struggle to identify root causes of previously unseen faults. This paper presents FaultExplainer, an interactive tool designed to improve fault detection, diagnosis, and explanation in the Tennessee Eastman Process (TEP). FaultExplainer integrates real-time sensor data visualization, Principal Component Analysis (PCA)-based fault detection, and identification of top contributing variables within an interactive user interface powered by large language models (LLMs). We evaluate the LLMs' reasoning capabilities in two scenarios: one where historical root causes are provided, and one where they are not to mimic the challenge of previously unseen faults. Experimental results using GPT-4o and o1-preview models demonstrate the system's strengths in generating plausible and actionable explanations, while also highlighting its limitations, including reliance on PCA-selected features and occasional hallucinations.

Stratospheric aerosols play an important role in the earth system and can affect the climate on timescales of months to years. However, estimating the characteristics of partially observed aerosol injections, such as those from volcanic eruptions, is fraught with uncertainties. This article presents a framework for stratospheric aerosol source inversion which accounts for background aerosol noise and earth system internal variability via a Bayesian approximation error approach. We leverage specially designed earth system model simulations using the Energy Exascale Earth System Model (E3SM). A comprehensive framework for data generation, data processing, dimension reduction, operator learning, and Bayesian inversion is presented where each component of the framework is designed to address particular challenges in stratospheric modeling on the global scale. We present numerical results using synthesized observational data to rigorously assess the ability of our approach to estimate aerosol sources and associate uncertainty with those estimates.

Schr\"{o}dinger bridge--a stochastic dynamical generalization of optimal mass transport--exhibits a learning-control duality. Viewed as a stochastic control problem, the Schr\"{o}dinger bridge finds an optimal control policy that steers a given joint state statistics to another while minimizing the total control effort subject to controlled diffusion and deadline constraints. Viewed as a stochastic learning problem, the Schr\"{o}dinger bridge finds the most-likely distribution-valued trajectory connecting endpoint distributional observations, i.e., solves the two point boundary-constrained maximum likelihood problem over the manifold of probability distributions. Recent works have shown that solving the Schr\"{o}dinger bridge problem with state cost requires finding the Markov kernel associated with a reaction-diffusion PDE where the state cost appears as a state-dependent reaction rate. We explain how ideas from Weyl calculus in quantum mechanics, specifically the Weyl operator and the Weyl symbol, can help determine such Markov kernels. We illustrate these ideas by explicitly finding the Markov kernel for the case of quadratic state cost via Weyl calculus, recovering our earlier results but avoiding tedious computation with Hermite polynomials.

Lambert's problem concerns with transferring a spacecraft from a given initial to a given terminal position within prescribed flight time via velocity control subject to a gravitational force field. We consider a probabilistic variant of the Lambert problem where the knowledge of the endpoint constraints in position vectors are replaced by the knowledge of their respective joint probability density functions. We show that the Lambert problem with endpoint joint probability density constraints is a generalized optimal mass transport (OMT) problem, thereby connecting this classical astrodynamics problem with a burgeoning area of research in modern stochastic control and stochastic machine learning. This newfound connection allows us to rigorously establish the existence and uniqueness of solution for the probabilistic Lambert problem. The same connection also helps to numerically solve the probabilistic Lambert problem via diffusion regularization, i.e., by leveraging further connection of the OMT with the Schr\"odinger bridge problem (SBP). This also shows that the probabilistic Lambert problem with additive dynamic process noise is in fact a generalized SBP, and can be solved numerically using the so-called Schr\"odinger factors, as we do in this work. We explain how the resulting analysis leads to solving a boundary-coupled system of reaction-diffusion PDEs where the nonlinear gravitational potential appears as the reaction rate. We propose novel algorithms for the same, and present illustrative numerical results. Our analysis and the algorithmic framework are nonparametric, i.e., we make neither statistical (e.g., Gaussian, first few moments, mixture or exponential family, finite dimensionality of the sufficient statistic) nor dynamical (e.g., Taylor series) approximations.

We perform a set of flow and reactive transport simulations within three-dimensional fracture networks to learn the factors controlling mineral reactions. CO$_2$ mineralization requires CO$_2$-laden water, dissolution of a mineral that then leads to precipitation of a CO$_2$-bearing mineral. Our discrete fracture networks (DFN) are partially filled with quartz that gradually dissolves until it reaches a quasi-steady state. At the end of the simulation, we measure the quartz remaining in each fracture within the domain. We observe that a small backbone of fracture exists, where the quartz is fully dissolved which leads to increased flow and transport. However, depending on the DFN topology and the rate of dissolution, we observe a large variability of these changes, which indicates an interplay between the fracture network structure and the impact of geochemical dissolution. In this work, we developed a machine learning framework to extract the important features that support mineralization in the form of dissolution. In addition, we use structural and topological features of the fracture network to predict the remaining quartz volume in quasi-steady state conditions. As a first step to characterizing carbon mineralization, we study dissolution with this framework. We studied a variety of reaction and fracture parameters and their impact on the dissolution of quartz in fracture networks. We found that the dissolution reaction rate constant of quartz and the distance to the flowing backbone in the fracture network are the two most important features that control the amount of quartz left in the system. For the first time, we use a combination of a finite-volume reservoir model and graph-based approach to study reactive transport in a complex fracture network to determine the key features that control dissolution.

Microkinetics allows detailed modelling of chemical transformations occurring in many industrially relevant reactions. Traditional way of solving the microkinetics model for Fischer-Tropsch synthesis (FTS) becomes inefficient when it comes to more advanced real-time applications. In this work, we address these challenges by using physics-informed neural networks(PINNs) for modelling FTS microkinetics. We propose a computationally efficient and accurate method, enabling the ultra-fast solution of the existing microkinetics models in realistic process conditions. The proposed PINN model computes the fraction of vacant catalytic sites, a key quantity in FTS microkinetics, with median relative error (MRE) of 0.03%, and the FTS product formation rates with MRE of 0.1%. Compared to conventional equation solvers, the model achieves up to 1E+06 times speed-up when running on GPUs, thus being fast enough for multi-scale and multi-physics reactor modelling and enabling its applications in real-time process control and optimization.

We investigate the dynamics of chemical reaction networks (CRNs) with the goal of deriving an upper bound on their reaction rates. This task is challenging due to the nonlinear nature and discrete structure inherent in CRNs. To address this, we employ an information geometric approach, using the natural gradient, to develop a nonlinear system that yields an upper bound for CRN dynamics. We validate our approach through numerical simulations, demonstrating faster convergence in a specific class of CRNs. This class is characterized by the number of chemicals, the maximum value of stoichiometric coefficients of the chemical reactions, and the number of reactions. We also compare our method to a conventional approach, showing that the latter cannot provide an upper bound on reaction rates of CRNs. While our study focuses on CRNs, the ubiquity of hypergraphs in fields from natural sciences to engineering suggests that our method may find broader applications, including in information science.

We propose a reinforcement learning based method to identify important configurations that connect reactant and product states along chemical reaction paths. By shooting multiple trajectories from these configurations, we can generate an ensemble of configurations that concentrate on the transition path ensemble. This configuration ensemble can be effectively employed in a neural network-based partial differential equation solver to obtain an approximation solution of a restricted Backward Kolmogorov equation, even when the dimension of the problem is very high. The resulting solution, known as the committor function, encodes mechanistic information for the reaction and can in turn be used to evaluate reaction rates.

Cataloging the complex behaviors of dynamical systems can be challenging, even when they are well-described by a simple mechanistic model. If such a system is of limited analytical tractability, brute force simulation is often the only resort. We present an alternative, optimization-driven approach using tools from machine learning. We apply this approach to a novel, fully-optimizable, reaction-diffusion model which incorporates complex chemical reaction networks (termed "Dense Reaction-Diffusion Network" or "Dense RDN"). This allows us to systematically identify new states and behaviors, including pattern formation, dissipation-maximizing nonequilibrium states, and replication-like dynamical structures.

A central object in the computational studies of rare events is the committor function. Though costly to compute, the committor function encodes complete mechanistic information of the processes involving rare events, including reaction rates and transition-state ensembles. Under the framework of transition path theory (TPT), recent work [1] proposes an algorithm where a feedback loop couples a neural network that models the committor function with importance sampling, mainly umbrella sampling, which collects data needed for adaptive training. In this work, we show additional modifications are needed to improve the accuracy of the algorithm. The first modification adds elements of supervised learning, which allows the neural network to improve its prediction by fitting to sample-mean estimates of committor values obtained from short molecular dynamics trajectories. The second modification replaces the committor-based umbrella sampling with the finite-temperature string (FTS) method, which enables homogeneous sampling in regions where transition pathways are located. We test our modifications on low-dimensional systems with non-convex potential energy where reference solutions can be found via analytical or the finite element methods, and show how combining supervised learning and the FTS method yields accurate computation of committor functions and reaction rates. We also provide an error analysis for algorithms that use the FTS method, using which reaction rates can be accurately estimated during training with a small number of samples. The methods are then applied to a molecular system in which no reference solution is known, where accurate computations of committor functions and reaction rates can still be obtained.