Luke Bhan

Miroslav Krstic, Luke Bhan, Yuanyuan Shi

Unlike ODEs, whose models involve system matrices and whose controllers involve vector or matrix gains, PDE models involve functions in those roles functional coefficients, dependent on the spatial variables, and gain functions dependent on space as well. The designs of gains for controllers and observers for PDEs, such as PDE backstepping, are mappings of system model functions into gain functions. These infinite dimensional nonlinear operators are given in an implicit form through PDEs, in spatial variables, which need to be solved to determine the gain function for each new functional coefficient of the PDE. The need for solving such PDEs can be eliminated by learning and approximating the said design mapping in the form of a neural operator. Learning the neural operator requires a sufficient number of prior solutions for the design PDEs, offline, as well as the training of the operator. In recent work, we developed the neural operators for PDE backstepping designs for first order hyperbolic PDEs. Here we extend this framework to the more complex class of parabolic PDEs. The key theoretical question is whether the controllers are still stabilizing, and whether the observers are still convergent, if they employ the approximate functional gains generated by the neural operator. We provide affirmative answers to these questions, namely, we prove stability in closed loop under gains produced by neural operators. We illustrate the theoretical results with numerical tests and publish our code on github. The neural operators are three orders of magnitude faster in generating gain functions than PDE solvers for such gain functions. This opens up the opportunity for the use of this neural operator methodology in adaptive control and in gain scheduling control for nonlinear PDEs.

Miroslav Krstic, Iasson Karafyllis, Luke Bhan et al.

Age-structured predator-prey integro-partial differential equations provide models of interacting populations in ecology, epidemiology, and biotechnology. A key challenge in feedback design for these systems is the scalar $ζ$, defined implicitly by the Lotka-Sharpe nonlinear integral condition, as a mapping from fertility and mortality rates to $ζ$. To solve this challenge with operator learning, we first prove that the Lotka-Sharpe operator is Lipschitz continuous, guaranteeing the existence of arbitrarily accurate neural operator approximations over a compact set of fertility and mortality functions. We then show that the resulting approximate feedback law preserves semi-global practical asymptotic stability under propagation of the operator approximation error through various other nonlinear operators, all the way through to the control input. In the numerical results, not only do we learn ``once-and-for-all'' the canonical Lotka-Sharpe (LS) operator, and thus make it available for future uses in control of other age-structured population interconnections, but we demonstrate the online usage of the neural LS operator under estimation of the fertility and mortality functions.

Luke Bhan, Marcos Quinones-Grueiro, Gautam Biswas

In this work, we address the problem of solving complex collaborative robotic tasks subject to multiple varying parameters. Our approach combines simultaneous policy blending with system identification to create generalized policies that are robust to changes in system parameters. We employ a blending network whose state space relies solely on parameter estimates from a system identification technique. As a result, this blending network learns how to handle parameter changes instead of trying to learn how to solve the task for a generalized parameter set simultaneously. We demonstrate our scheme's ability on a collaborative robot and human itching task in which the human has motor impairments. We then showcase our approach's efficiency with a variety of system identification techniques when compared to standard domain randomization.

Luke Bhan, Yuanyuan Shi, Miroslav Krstic

Neural operator approximations of the gain kernels in PDE backstepping has emerged as a viable method for implementing controllers in real time. With such an approach, one approximates the gain kernel, which maps the plant coefficient into the solution of a PDE, with a neural operator. It is in adaptive control that the benefit of the neural operator is realized, as the kernel PDE solution needs to be computed online, for every updated estimate of the plant coefficient. We extend the neural operator methodology from adaptive control of a hyperbolic PDE to adaptive control of a benchmark parabolic PDE (a reaction-diffusion equation with a spatially-varying and unknown reaction coefficient). We prove global stability and asymptotic regulation of the plant state for a Lyapunov design of parameter adaptation. The key technical challenge of the result is handling the 2D nature of the gain kernels and proving that the target system with two distinct sources of perturbation terms, due to the parameter estimation error and due to the neural approximation error, is Lyapunov stable. To verify our theoretical result, we present simulations achieving calculation speedups up to 45x relative to the traditional finite difference solvers for every timestep in the simulation trajectory.

Maxence Lamarque, Luke Bhan, Yuanyuan Shi et al.

To stabilize PDEs, feedback controllers require gain kernel functions, which are themselves governed by PDEs. Furthermore, these gain-kernel PDEs depend on the PDE plants' functional coefficients. The functional coefficients in PDE plants are often unknown. This requires an adaptive approach to PDE control, i.e., an estimation of the plant coefficients conducted concurrently with control, where a separate PDE for the gain kernel must be solved at each timestep upon the update in the plant coefficient function estimate. Solving a PDE at each timestep is computationally expensive and a barrier to the implementation of real-time adaptive control of PDEs. Recently, results in neural operator (NO) approximations of functional mappings have been introduced into PDE control, for replacing the computation of the gain kernel with a neural network that is trained, once offline, and reused in real-time for rapid solution of the PDEs. In this paper, we present the first result on applying NOs in adaptive PDE control, presented for a benchmark 1-D hyperbolic PDE with recirculation. We establish global stabilization via Lyapunov analysis, in the plant and parameter error states, and also present an alternative approach, via passive identifiers, which avoids the strong assumptions on kernel differentiability. We then present numerical simulations demonstrating stability and observe speedups up to three orders of magnitude, highlighting the real-time efficacy of neural operators in adaptive control. Our code (Github) is made publicly available for future researchers.

Luke Bhan, Yuexin Bian, Miroslav Krstic et al.

Over the last decade, data-driven methods have surged in popularity, emerging as valuable tools for control theory. As such, neural network approximations of control feedback laws, system dynamics, and even Lyapunov functions have attracted growing attention. With the ascent of learning based control, the need for accurate, fast, and easy-to-use benchmarks has increased. In this work, we present the first learning-based environment for boundary control of PDEs. In our benchmark, we introduce three foundational PDE problems - a 1D transport PDE, a 1D reaction-diffusion PDE, and a 2D Navier-Stokes PDE - whose solvers are bundled in an user-friendly reinforcement learning gym. With this gym, we then present the first set of model-free, reinforcement learning algorithms for solving this series of benchmark problems, achieving stability, although at a higher cost compared to model-based PDE backstepping. With the set of benchmark environments and detailed examples, this work significantly lowers the barrier to entry for learning-based PDE control - a topic largely unexplored by the data-driven control community. The entire benchmark is available on Github along with detailed documentation and the presented reinforcement learning models are open sourced.

Luke Bhan, Miroslav Krstic, Yuanyuan Shi

Due to simplicity and strong stability guarantees, predictor feedback methods have stood as a popular approach for time delay systems since the 1950s. For time-varying delays, however, implementation requires computing a prediction horizon defined by the inverse of the delay function, which is rarely available in closed form and must be approximated. In this work, we formulate the inverse delay mapping as an operator learning problem and study predictor feedback under approximation of the prediction horizon. We propose two approaches: (i) a numerical method based on time integration of an equivalent ODE, and (ii) a data-driven method using neural operators to learn the inverse mapping. We show that both approaches achieve arbitrary approximation accuracy over compact sets, with complementary trade-offs in computational cost and scalability. Building on these approximations, we then develop an output-feedback predictor design for systems with delays in both the input and the measurement. We prove that the resulting closed-loop system is globally exponentially stable when the prediction horizon is approximated with sufficiently small error. Lastly, numerical experiments validate the proposed methods and illustrate their trade-offs between accuracy and computational efficiency.

Luke Bhan, Peter Quawas, Miroslav Krstic et al.

Modern control systems frequently operate under input delays and sampled state measurements. A common delay-compensation strategy is predictor feedback; however, practical implementations require solving an implicit ODE online, resulting in intractable computational cost. Moreover, predictor formulations typically assume continuously available state measurements, whereas in practice measurements may be sampled, irregular, or temporarily missing due to hardware faults. In this work, we develop two neural-operator predictor-feedback designs for nonlinear systems with delayed inputs and sampled measurements. In the first design, we introduce a sampling-horizon prediction operator that maps the current measurement and input history to the predicted state trajectory over the next sampling interval. In the second design, the neural operator approximates only the delay-compensating predictor, which is then composed with the closed-loop flow between measurements. The first approach requires uniform sampling but yields residual bounds that scale directly with the operator approximation error. In contrast, the second accommodates non-uniform, but bounded sampling schedules at the cost of amplified approximation error, revealing a practical tradeoff between sampling flexibility and approximation sensitivity for the control engineer. For both schemes, we establish semi-global practical stability with explicit neural operator error-dependent bounds. Numerical experiments on a 6-link nonlinear robotic manipulator demonstrate accurate tracking and substantial computational speedup of 25$\times$ over a baseline approach.

Luke Bhan, Hanyu Zhang, Andrew Gordon Wilson et al.

Monitoring forecasting systems is critical for customer satisfaction, profitability, and operational efficiency in large-scale retail businesses. We propose The Forecast Critic, a system that leverages Large Language Models (LLMs) for automated forecast monitoring, taking advantage of their broad world knowledge and strong ``reasoning'' capabilities. As a prerequisite for this, we systematically evaluate the ability of LLMs to assess time series forecast quality, focusing on three key questions. (1) Can LLMs be deployed to perform forecast monitoring and identify obviously unreasonable forecasts? (2) Can LLMs effectively incorporate unstructured exogenous features to assess what a reasonable forecast looks like? (3) How does performance vary across model sizes and reasoning capabilities, measured across state-of-the-art LLMs? We present three experiments, including on both synthetic and real-world forecasting data. Our results show that LLMs can reliably detect and critique poor forecasts, such as those plagued by temporal misalignment, trend inconsistencies, and spike errors. The best-performing model we evaluated achieves an F1 score of 0.88, somewhat below human-level performance (F1 score: 0.97). We also demonstrate that multi-modal LLMs can effectively incorporate unstructured contextual signals to refine their assessment of the forecast. Models correctly identify missing or spurious promotional spikes when provided with historical context about past promotions (F1 score: 0.84). Lastly, we demonstrate that these techniques succeed in identifying inaccurate forecasts on the real-world M5 time series dataset, with unreasonable forecasts having an sCRPS at least 10% higher than that of reasonable forecasts. These findings suggest that LLMs, even without domain-specific fine-tuning, may provide a viable and scalable option for automated forecast monitoring and evaluation.

Maxence Lamarque, Luke Bhan, Rafael Vazquez et al.

To stabilize PDE models, control laws require space-dependent functional gains mapped by nonlinear operators from the PDE functional coefficients. When a PDE is nonlinear and its "pseudo-coefficient" functions are state-dependent, a gain-scheduling (GS) nonlinear design is the simplest approach to the design of nonlinear feedback. The GS version of PDE backstepping employs gains obtained by solving a PDE at each value of the state. Performing such PDE computations in real time may be prohibitive. The recently introduced neural operators (NO) can be trained to produce the gain functions, rapidly in real time, for each state value, without requiring a PDE solution. In this paper we introduce NOs for GS-PDE backstepping. GS controllers act on the premise that the state change is slow and, as a result, guarantee only local stability, even for ODEs. We establish local stabilization of hyperbolic PDEs with nonlinear recirculation using both a "full-kernel" approach and the "gain-only" approach to gain operator approximation. Numerical simulations illustrate stabilization and demonstrate speedup by three orders of magnitude over traditional PDE gain-scheduling. Code (Github) for the numerical implementation is published to enable exploration.

Luke Bhan, Peijia Qin, Miroslav Krstic et al.

Predictor feedback designs are critical for delay-compensating controllers in nonlinear systems. However, these designs are limited in practical applications as predictors cannot be directly implemented, but require numerical approximation schemes, which become computationally prohibitive when system dynamics are expensive to compute. To address this challenge, we recast the predictor design as an operator learning problem, and learn the predictor mapping via a neural operator. We prove the existence of an arbitrarily accurate neural operator approximation of the predictor operator. Under the approximated predictor, we achieve semiglobal practical stability of the closed-loop nonlinear delay system. The estimate is semiglobal in a unique sense - one can enlarge the set of initial states as desired, though this increases the difficulty of training a neural operator, which appears practically in the stability estimate. Furthermore, our analysis holds for any black-box predictor satisfying the universal approximation error bound. We demonstrate the approach by controlling a 5-link robotic manipulator with different neural operator models, achieving significant speedups compared to classic predictor feedback schemes while maintaining closed-loop stability.

Luke Bhan, Miroslav Krstic, Yuanyuan Shi

In this work, we propose a rigorous method for implementing predictor feedback controllers in nonlinear systems with unknown and arbitrarily long actuator delays. To address the analytically intractable nature of the predictor, we approximate it using a learned neural operator mapping. This mapping is trained once, offline, and then deployed online, leveraging the fast inference capabilities of neural networks. We provide a theoretical stability analysis based on the universal approximation theorem of neural operators and the transport partial differential equation (PDE) representation of the delay. We then prove, via a Lyapunov-Krasovskii functional, semi-global practical convergence of the dynamical system dependent on the approximation error of the predictor and delay bounds. Finally, we validate our theoretical results using a biological activator/repressor system, demonstrating speedups of 15 times compared to traditional numerical methods.

Luke Bhan, Yuanyuan Shi, Iasson Karafyllis et al.

Observers for PDEs are themselves PDEs. Therefore, producing real time estimates with such observers is computationally burdensome. For both finite-dimensional and ODE systems, moving-horizon estimators (MHE) are operators whose output is the state estimate, while their inputs are the initial state estimate at the beginning of the horizon as well as the measured output and input signals over the moving time horizon. In this paper we introduce MHEs for PDEs which remove the need for a numerical solution of an observer PDE in real time. We accomplish this using the PDE backstepping method which, for certain classes of both hyperbolic and parabolic PDEs, produces moving-horizon state estimates explicitly. Precisely, to explicitly produce the state estimates, we employ a backstepping transformation of a hard-to-solve observer PDE into a target observer PDE, which is explicitly solvable. The MHEs we propose are not new observer designs but simply the explicit MHE realizations, over a moving horizon of arbitrary length, of the existing backstepping observers. Our PDE MHEs lack the optimality of the MHEs that arose as duals of MPC, but they are given explicitly, even for PDEs. In the paper we provide explicit formulae for MHEs for both hyperbolic and parabolic PDEs, as well as simulation results that illustrate theoretically guaranteed convergence of the MHEs.

Luke Bhan, Miroslav Krstic, Yuanyuan Shi

This work establishes the first rigorous stability guarantees for approximate predictors in delay-adaptive control of nonlinear systems, addressing a key challenge in practical implementations where exact predictors are unavailable. We analyze two scenarios: (i) when the actuated input is directly measurable, and (ii) when it is estimated online. For the measurable input case, we prove semi-global practical asymptotic stability with an explicit bound proportional to the approximation error $ε$. For the unmeasured input case, we demonstrate local practical asymptotic stability, with the region of attraction explicitly dependent on both the initial delay estimate and the predictor approximation error. To bridge theory and practice, we show that neural operators-a flexible class of neural network-based approximators-can achieve arbitrarily small approximation errors, thus satisfying the conditions of our stability theorems. Numerical experiments on two nonlinear benchmark systems-a biological protein activator/repressor model and a micro-organism growth Chemostat model-validate our theoretical results. In particular, our numerical simulations confirm stability under approximate predictors, highlight the strong generalization capabilities of neural operators, and demonstrate a substantial computational speedup of up to 15x compared to a baseline fixed-point method.

Filip Bajraktari, Luke Bhan, Miroslav Krstic et al.

In this work, we present the first stability results for approximate predictors in multi-input non-linear systems with distinct actuation delays. We show that if the predictor approximation satisfies a uniform (in time) error bound, semi-global practical stability is correspondingly achieved. For such approximators, the required uniform error bound depends on the desired region of attraction and the number of control inputs in the system. The result is achieved through transforming the delay into a transport PDE and conducting analysis on the coupled ODE-PDE cascade. To highlight the viability of such error bounds, we demonstrate our results on a class of approximators - neural operators - showcasing sufficiency for satisfying such a universal bound both theoretically and in simulation on a mobile robot experiment.

Adam Stager, Luke Bhan, Andreas Malikopoulos et al.

The common thread that characterizes energy efficient mobility systems for smart cities is their interconnectivity which enables the exchange of massive amounts of data; this, in turn, provides the opportunity to develop a decentralized framework to process this information and deliver real-time control actions that optimize energy consumption and other associated benefits. To seize these opportunities, this paper describes the development of a scaled smart city providing a glimpse that bridges the gap between simulation and full scale implementation of energy efficient mobility systems. Using this testbed, we can quickly, safely, and affordably experimentally validate control concepts aimed at enhancing our understanding of the implications of next generation mobility systems.