Research InterestsA (Hopefully) Easy Introduction to Our ResearchOur group develops mathematical and computational tools – rooted in systems theory, operator theory, and machine learning – for modeling, analyzing, and controlling complex chemical processes and generally any dynamical systems. Th subsections below aims to give a gentle, non-technical entry point into what we do and why it matters, aimed at anyone with a standard chemical engineering background. Credits should be given to Claude AI for generating these. Introduction to process systems engineering for chemical engineersA single unit operation – a reactor, a distillation column, a heat exchanger – is already familiar from your core chemical engineering courses: write mass and energy balances, maybe add a simple feedback controller, and you can predict and regulate its behavior. Process systems engineering (PSE) asks a harder question: what happens when dozens or hundreds of such units are connected into a plant, with material and energy recycles, shared utilities, and competing objectives like yield, energy efficiency, safety, and environmental impact? At this scale, the plant behaves as one large, interconnected dynamical system, and decisions made in one unit can propagate – often with delay and in nonlinear ways – to affect every other unit downstream and upstream. PSE provides the modeling, simulation, optimization, and control theory needed to design, operate, and coordinate such systems as a whole, rather than one unit at a time. Our group's research sits inside this tradition, but focuses on plants and processes whose behavior is too complex, too nonlinear, or too poorly characterized for classical first-principles models alone – which is where data-driven methods, described next, come in. Machine learning and data-driven controlMany processes of practical interest – multiphase reactors, biological systems, novel separation processes – are difficult to model accurately from first principles alone. Deriving a trustworthy set of governing equations can require years of mechanistic study, and the resulting model may still be too complex to use directly for real-time control or optimization. An alternative is to let data do some of the work: use measurements collected from the process (sensor readings, historical operating data, or designed experiments) to learn a model of its dynamics, or even to learn a control policy directly, without fully deriving the underlying physics. This is the essence of data-driven control, a natural marriage of machine learning with classical process control and estimation theory. Our group's philosophy is that data-driven methods should not be used as a black box that replaces engineering understanding, but as a way to responsibly extend it: our data-driven models and controllers are built to respect known physical structure (such as stability, safety constraints, and steady-state behavior) and to come with rigorous theoretical guarantees, so that they can be trusted and certified the way a first-principles model would be, while remaining applicable to processes that are otherwise too complex to model by hand. Koopman operator theoryMost chemical processes are nonlinear: double the flow rate into a reactor and the outlet concentration typically does not simply double in response. Nonlinear systems are, in general, far harder to analyze and control than linear ones, for which control engineers have a century of mature and well-understood theory (stability analysis, optimal control, filtering, and so on). The Koopman operator offers an elegant way to bring that linear toolbox back into play. Instead of tracking how the state of a nonlinear system evolves in time, the Koopman operator tracks how functions of the state – called observables – evolve. Remarkably, this shift in perspective turns a nonlinear problem into a linear one: the Koopman operator itself is linear, even though it acts on an infinite-dimensional space of observable functions and exactly describes the original nonlinear dynamics. In practice, we approximate this operator from data using a finite set of chosen or learned observables, which yields a linear surrogate model of the nonlinear process. A substantial part of our group's research is dedicated to making this approximation both accurate and trustworthy: understanding when and how well it converges to the true operator, using it to identify unknown system dynamics, to estimate states that cannot be measured directly, and to analyze and certify stability, all while keeping the resulting tools practical for real chemical engineering systems. Efficient optimization algorithms for process systems engineering and controlUnderneath almost every design, operation, and control decision in process systems engineering lies an optimization problem: choose the flow rates, temperatures, or controller actions that minimize cost or maximize yield, subject to safety and equipment constraints. As models grow more detailed and plants grow more interconnected, these optimization problems grow too – easily into thousands or millions of decision variables when a full plantwide schedule, a distributed control system, or a fine-grained physical model is involved. Solving such high-dimensional optimization problems reliably, and doing so fast enough to be useful in real time, is itself a major research challenge, and a recurring theme in our group's work on distributed and large-scale optimization algorithms. A related challenge arises when the objective or constraints of an optimization problem cannot be written down explicitly – for instance, when evaluating them requires running an expensive simulation or a physical experiment, and no clean formula or gradient is available. This is the setting of black-box optimization, where an algorithm must decide what to try next based only on the outcomes of previous trials. Efficient algorithms of this kind have applications well beyond conventional process optimization: in molecular simulation, they can guide the exploration of a molecule's conformational or reaction landscape without expensive first-principles calculations at every step, and in AI-driven molecular discovery, they underlie how generative and search algorithms navigate the enormous space of candidate molecules or materials to identify promising ones for further study. Our group is interested in bringing the same rigor we apply to process optimization – convergence guarantees, scalability, and robustness to noise – to these emerging, high-dimensional and black-box optimization problems. Why advanced mathematics matters, and why you should not be afraid of itMuch of the theory described above – the convergence of data-driven Koopman approximations, the guarantees behind data-driven control, the behavior of high-dimensional optimization algorithms – ultimately rests on mathematical tools that go beyond the calculus and linear algebra taught in a standard chemical engineering curriculum: real analysis, which makes rigorous the notions of limits, continuity, and convergence that underlie any claim that an algorithm or approximation “works,” and functional analysis, which extends linear algebra from finite-dimensional vectors to infinite-dimensional spaces of functions and operators, exactly the setting in which the Koopman operator lives. Without this language, it is difficult to state precisely what it even means for a data-driven model to be accurate or a controller to be safe, let alone to prove it. That said, chemical engineers are better positioned to pick up this material than they often expect. The core ideas of real and functional analysis are, in a real sense, the rigorous completion of concepts already familiar from a chemical engineering education: the epsilon-delta definitions of real analysis formalize the same intuitions about limits and continuity used in transport phenomena and process dynamics, while functional analysis generalizes the linear algebra used in solving balance equations and control systems to the function spaces that show up naturally in distributed parameter systems, spectral methods, and now data-driven modeling. Our own experience is that a willingness to sit with the definitions, work through a few proofs, and ask “why” rather than only “how” is far more important than any specific prior coursework. We view learning this mathematics not as an obstacle guarding the field, but as a genuinely rewarding extension of the engineering intuition chemical engineers already have – and one of the most useful investments a student in our group can make. Research Funding
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