Clarence W. Rowley

Abstract Temporal gradient sensing is a fundamental capability observed across diverse natural biological systems, contributing to the coordination of their functions. Harnessing this ability is also of significant interest in synthetic biology, particularly for sensing and control applications. In this work, we focus on a biomolecular topology that exemplifies a broader class of signal-differentiating architectures, while introducing a structural variant of it. We examine their behavior under both nominal and non-ideal conditions, accounting for stochastic noise arising from different sources. Our investigation includes scenarios where these topologies operate independently, as well as when embedded within minimal regulatory architectures based on negative as well as positive feedback. We analyze the stability of the resulting macroscopic dynamics—a prerequisite for practical deployment—and quantify stochastic fluctuations in system output, providing comparisons with the corresponding input/unregulated process. Importantly, our results demonstrate that signal differentiation can be effectively implemented in a biomolecular setting without incurring deleterious noise amplification—a major concern in the utilization of derivative action across disciplines.

Abstract Achieving complex multi-species control objectives is essential for engineering advanced autoregulated biomolecular devices. This paper addresses the problem of robust steady-state tracking for outputs defined as multiplicative combinations of biomolecular species concentrations. We first introduce a control architecture realized via chemical reaction networks that steers the product of two target species concentrations in the controlled network to a prescribed value. A robust stability analysis is provided for closed-loop system families with distinct structural characteristics. The proposed framework is also extended to a more general formulation capable of regulating arbitrary monomial outputs involving multiple species. Numerical simulations of representative examples corroborate the theoretical results and illustrate the effectiveness of our approach.

Achieving complex multi-species control objectives is essential for engineering advanced autoregulated biomolecular devices. This paper addresses the problem of robust steady-state tracking for outputs defined as multiplicative combinations of biomolecular species concentrations. We first introduce a control architecture realized via chemical reaction networks that steers the product of two target species concentrations in the controlled network to a prescribed value. A robust stability analysis is provided for closed-loop system families with distinct structural characteristics. The proposed framework is also extended to a more general formulation capable of regulating arbitrary monomial outputs involving multiple species. Numerical simulations of representative examples corroborate the theoretical results and illustrate the effectiveness of our approach.

In this paper, we present resolvent4py , a parallel Python package for the analysis, model reduction and control of large-scale linear systems with millions or billions of degrees of freedom. This package provides the user with a friendly Python-like experience (akin to that of well-established libraries such as numpy and scipy ), while enabling MPI-based parallelism through mpi4py , petsc4py and slepc4py . In turn, this allows for the development of streamlined and efficient Python code that can be used to solve several problems in fluid mechanics, solid mechanics, graph theory, molecular dynamics and several other fields.

We consider data-driven reduced-order models of partial differential equations with shift equivariance. Shift-equivariant systems typically admit traveling solutions, and the main idea of our approach is to represent the solution in a traveling reference frame, in which it can be described by a relatively small number of basis functions. Existing methods for operator inference allow one to approximate a reduced-order model directly from data, without knowledge of the full-order dynamics. Our method adds additional terms to ensure that the reduced-order model not only approximates the spatially frozen profile of the solution, but also estimates the traveling speed as a function of that profile. We validate our approach using the Kuramoto-Sivashinsky equation, a one-dimensional partial differential equation that exhibits traveling solutions and spatiotemporal chaos. Results indicate that our method robustly captures traveling solutions, and exhibits improved numerical stability over the standard operator inference approach.

In this paper, we present resolvent4py, a parallel Python package for the analysis, model reduction and control of large-scale linear systems with millions or billions of degrees of freedom. This package provides the user with a friendly Python-like experience (akin to that of well-established libraries such as numpy and scipy), while enabling MPI-based parallelism through mpi4py, petsc4py and slepc4py. In turn, this allows for the development of streamlined and efficient Python code that can be used to solve several problems in fluid mechanics, solid mechanics, graph theory, molecular dynamics and several other fields.

With the rapid increase in data resources and computational power as well as the accompanying current trend to incorporate machine learning into existing methods, data-driven approaches for modelling, analysis, and control of dynamical systems have attracted new interest and opened doors to novel applications. However, there is always a discrepancy between mathematical models and reality such that rigorously-shown error bounds and uncertainty quantification are indispensable for a reliable use of data-driven techniques, e.g., using surrogate models in optimisation-based control. Similar comments apply to data-enhanced models. Consequently, uncertainty about parameters, the model itself and numerous other aspects need to be taken into account, e.g., in data-driven control of (stochastic) dynamical systems. Hence, the respective paradigm changes have led to a variety of novel concepts which, however, still suffer from limitations: many concentrate only on a single aspect, are only applicable to systems of limited complexity, or lack a sound mathematical foundation including guarantees on feasibility, robustness, or the overall performance. Pushing these limits, we face a wide spectrum of theoretic and algorithmic challenges in modeling, analysis, and control under uncertainty using data-driven methods.

Here and in a companion paper, we consider a simple control problem in which the underlying dynamics depend on a parameter a that is unknown and must be learned. In this paper, we assume that a can be any real number and we do not assume that we have a prior belief about a . We seek a control strategy that minimizes a quantity called the regret. Given any \varepsilon>0 , we produce a strategy that minimizes the regret to within a multiplicative factor of (1+\varepsilon) .

Here and in a follow-on paper, we consider a simple control problem in which the underlying dynamics depend on a parameter a that is unknown and must be learned. In this paper, we assume that a is bounded, i.e., that |a| \le a_{\textrm{MAX}} , and we study two variants of the control problem. In the first variant, Bayesian control, we are given a prior probability distribution for a and we seek a strategy that minimizes the expected value of a given cost function. Assuming that we can solve a certain PDE (the Hamilton–Jacobi–Bellman equation), we produce optimal strategies for Bayesian control. In the second variant, agnostic control, we assume nothing about a and we seek a strategy that minimizes a quantity called the regret. We produce a prior probability distribution d\textup{Prior}(a) supported on a finite subset of [-a_{\textrm{MAX}},a_{\textrm{MAX}}] so that the agnostic control problem reduces to the Bayesian control problem for the prior d\textup{Prior}(a) .

.Data-driven models for nonlinear dynamical systems based on approximating the underlying Koopman operator or generator have proven to be successful tools for forecasting, feature learning, state estimation, and control. It has become well known that the Koopman generators for control-affine systems also have affine dependence on the input, leading to convenient finite-dimensional bilinear approximations of the dynamics. Yet there are still two main obstacles that limit the scope of current approaches for approximating the Koopman generators of systems with actuation. First, the performance of existing methods depends heavily on the choice of basis functions over which the Koopman generator is to be approximated; and there is currently no universal way to choose them for systems that are not measure preserving. Second, if we do not observe the full state, then it becomes necessary to account for the dependence of the output time series on the sequence of supplied inputs when constructing observables to approximate Koopman operators. To address these issues, we write the dynamics of observables governed by the Koopman generator as a bilinear hidden Markov model and determine the model parameters using the expectation-maximization algorithm. The E step involves a standard Kalman filter and smoother, while the M step resembles control-affine dynamic mode decomposition for the generator. We demonstrate the performance of this method on three examples, including recovery of a finite-dimensional Koopman-invariant subspace for an actuated system with a slow manifold; estimation of Koopman eigenfunctions for the unforced Duffing equation; and model-predictive control of a fluidic pinball system based only on noisy observations of lift and drag.KeywordsKoopman generatorcontrol-affine systembilinear systemhidden Markov modelexpectation-maximization algorithmdata-driven system identificationMSC codes37A5037C3037C6037M1037N1037N3547D0347D0660G3560J0562M0562M1562M2093B2893B3093B4593E1093E1193E1293E14

We consider a simple control problem in which the underlying dynamics depend on a parameter $a$ that is unknown and must be learned. We study three variants of the control problem: Bayesian control, in which we have a prior belief about $a$; bounded agnostic control, in which we have no prior belief about $a$ but we assume that $a$ belongs to a bounded set; and fully agnostic control, in which $a$ is allowed to be an arbitrary real number about which we have no prior belief. In the Bayesian variant, a control strategy is optimal if it minimizes a certain expected cost. In the agnostic variants, a control strategy is optimal if it minimizes a quantity called the worst-case regret. For the Bayesian and bounded agnostic variants above, we produce optimal control strategies. For the fully agnostic variant, we produce almost optimal control strategies, i.e., for any $\varepsilon>0$ we produce a strategy that minimizes the worst-case regret to within a multiplicative factor of $(1+\varepsilon)$.

Recently developed reduced-order modeling techniques aim to approximate nonlinear dynamical systems on low-dimensional manifolds learned from data. This is an effective approach for modeling dynamics in a post-transient regime where the effects of initial conditions and other disturbances have decayed. However, modeling transient dynamics near an underlying manifold, as needed for real-time control and forecasting applications, is complicated by the effects of fast dynamics and nonnormal sensitivity mechanisms. To begin to address these issues, we introduce a parametric class of nonlinear projections described by constrained autoencoder neural networks in which both the manifold and the projection fibers are learned from data. Our architecture uses invertible activation functions and biorthogonal weight matrices to ensure that the encoder is a left inverse of the decoder. We also introduce new dynamics-aware cost functions that promote learning of oblique projection fibers that account for fast dynamics and nonnormality. To demonstrate these methods and the specific challenges they address, we provide a detailed case study of a three-state model of vortex shedding in the wake of a bluff body immersed in a fluid, which has a two-dimensional slow manifold that can be computed analytically. In anticipation of future applications to high-dimensional systems, we also propose several techniques for constructing computationally efficient reduced-order models using our proposed nonlinear projection framework. This includes a novel sparsity-promoting penalty for the encoder that avoids detrimental weight matrix shrinkage via computation on the Grassmann manifold.

Reduced-order models for flows that exhibit time-periodic behavior are critical for several tasks, including active control and optimization. One well-known procedure to obtain the desired reduced-order model in the proximity of a periodic solution of the governing equations is continuous-time balanced truncation. Within this framework, the periodic reachability and observability Gramians are usually estimated numerically via quadrature using the forward and adjoint post-transient response to impulses. However, this procedure can be computationally expensive, especially in the presence of slowly-decaying transients. Moreover, it can only be performed if the periodic orbit is stable in the sense of Floquet. In order to address these issues, we use the frequency-domain representation of the Gramians, which we henceforth refer to as frequential Gramians. First, these frequential Gramians are well-defined for both stable and unstable dynamics. In particular, we show that when the underlying system is unstable, these Gramians satisfy a pair of allied differential Lyapunov equations. Second, they can be estimated numerically by solving algebraic systems of equations that lend themselves to heavy computational parallelism and that deliver the desired post-transient response without having to follow physical transients. We demonstrate the method on a periodically-forced axisymmetric jet at Reynolds numbers Re=1250 and Re=1500. At the lower Reynolds number, the flow strongly amplifies subharmonic perturbations and exhibits vortex pairing about a Floquet-stable T-periodic solution. At the higher Reynolds number, the underlying T-periodic orbit is unstable and the flow naturally settles onto a 2T-periodic limit cycle characterized by pairing vortices. At both Reynolds numbers, we use a balanced reduced-order model to design a feedback controller and a state estimator to suppress vortex pairing.

Here and in a follow-on paper, we consider a simple control problem in which the underlying dynamics depend on a parameter $a$ that is unknown and must be learned. In this paper, we assume that $a$ is bounded, i.e., that $|a| \le a_{\text{MAX}}$, and we study two variants of the control problem. In the first variant, Bayesian control, we are given a prior probability distribution for $a$ and we seek a strategy that minimizes the expected value of a given cost function. Assuming that we can solve a certain PDE (the Hamilton-Jacobi-Bellman equation), we produce optimal strategies for Bayesian control. In the second variant, agnostic control, we assume nothing about $a$ and we seek a strategy that minimizes a quantity called the regret. We produce a prior probability distribution $d\text{Prior}(a)$ supported on a finite subset of $[-a_{\text{MAX}},a_{\text{MAX}}]$ so that the agnostic control problem reduces to the Bayesian control problem for the prior $d\text{Prior}(a)$.

Here and in a companion paper, we consider a simple control problem in which the underlying dynamics depend on a parameter $a$ that is unknown and must be learned. In this paper, we assume that $a$ can be any real number and we do not assume that we have a prior belief about $a$. We seek a control strategy that minimizes a quantity called the regret. Given any $\varepsilon>0$, we produce a strategy that minimizes the regret to within a multiplicative factor of $(1+\varepsilon)$.

Recently developed reduced-order modeling techniques aim to approximate nonlinear dynamical systems on low-dimensional manifolds learned from data. This is an effective approach for modeling dynamics in a post-transient regime where the effects of initial conditions and other disturbances have decayed. However, modeling transient dynamics near an underlying manifold, as needed for real-time control and forecasting applications, is complicated by the effects of fast dynamics and nonnormal sensitivity mechanisms. To begin to address these issues, we introduce a parametric class of nonlinear projections described by constrained autoencoder neural networks in which both the manifold and the projection fibers are learned from data. Our architecture uses invertible activation functions and biorthogonal weight matrices to ensure that the encoder is a left inverse of the decoder. We also introduce new dynamics-aware cost functions that promote learning of oblique projection fibers that account for fast dynamics and nonnormality. To demonstrate these methods and the specific challenges they address, we provide a detailed case study of a three-state model of vortex shedding in the wake of a bluff body immersed in a fluid, which has a two-dimensional slow manifold that can be computed analytically. In anticipation of future applications to high-dimensional systems, we also propose several techniques for constructing computationally efficient reduced-order models using our proposed nonlinear projection framework. This includes a novel sparsity-promoting penalty for the encoder that avoids detrimental weight matrix shrinkage via computation on the Grassmann manifold.