Alexander Leonid Heide

2026

Data-driven nonlinear aerodynamics models with certifiably optimal boundedness properties

A. Leonid Heide, Shih-Chi Liao, Sergio Castiblanco-Ballesteros, Gustaaf B. Jacobs, Peter Seiler, Maziar S. Hemati

Journal of Fluid Mechanics Journal JFM

Obtaining predictive low-order models is a central challenge in fluid dynamics. Data-driven frameworks have been widely used to obtain low-order models of aerodynamic systems; yet, resulting models tend to yield predictions that grow unbounded with time. Recently introduced stability-promoting methods can facilitate the identification of bounded models, but tend to require extensive brute-force tuning even in the context of simple academic systems. Here, we show how recent theoretical advances in the long-term boundedness of dynamical systems can be integrated into data-driven modeling frameworks to ensure that resulting models will yield bounded predictions of incompressible flows. Specifically, we propose to solve a particular set of convex semidefinite programming problems to certify whether a system admits a globally attracting bounded set for the chosen modeling parameters and compute a model with the optimal tightest bound on this globally attracting set. We demonstrate the approach via integration within the sparse identification of nonlinear dynamics modeling framework. Application on two low-order benchmark problems establishes the merits of the approach. We then apply our approach to obtain a low-order 6-mode model of unsteady separation over a NACA-65(1)-412 airfoil at Re = 20,000, a flow that has been notoriously difficult to model using data-driven methods. The resulting model accurately predicts the dynamics of unsteady separation, with model predictions remaining bounded indefinitely. We anticipate this work will benefit future efforts in modeling strongly nonlinear flows, especially in settings where physically viable long-term forecasts are paramount.

The Problem: Data-driven models of fluid flows often work well for short-term predictions but eventually diverge or predict physically impossible behavior over long periods. Without explicit constraints, standard modeling algorithms tend to produce equations that are mathematically and physically inconsistent with the laws of fluid dynamics.
The Goal: To combine data-driven modeling with optimization methods to build simplified, nonlinear models that are mathematically guaranteed to stay bounded indefinitely, while maintaining a structure consistent with the governing fluid equations. In other words, we want to build a model that captures the behaviors present in the data while being (provably) physically accurate.
The Outcome: We developed a framework that leverages our “Trapping-SDP” approach to enforce strict physical bounds on a model’s long-term behavior. We proved its success on a complex, separated airflow over an airfoil, a scenario where standard learned models almost always fail.

Practical Airflow Modeling and Control from Data: A Physics-Informed Approach

A. Leonid Heide

Ph.D. Dissertation, University of Minnesota Ph.D. Dissertation

Paper

Unsteady aerodynamic phenomena, such as flow separation, degrade aircraft performance and necessitate flow control strategies to ensure safe and efficient flight. However, the real-time implementation of active flow control is bottlenecked by a lack of computationally inexpensive, reliable models. High-fidelity methods such as computational fluid dynamics (CFD) accurately capture first-principles physics but are prohibitively expensive for control design. Conversely, data-driven reduced-order models (ROMs) attempt to bridge this gap by fitting low-dimensional state equations to flow snapshots. Without explicit constraints, however, these empirical fits can violate the fundamental physical properties of fluids---such as lossless nonlinear energy exchange---leading to nonphysical long-time growth and unreliable predictions. The objective of this dissertation is therefore to create effective methods for modeling and controlling complex airflows by leveraging real-world data while respecting the governing fluid mechanics. Specifically, this work argues that respecting the physics when designing modeling and control strategies yields practical tools for real-time prediction and control. To achieve this, the dissertation advances four methodologies. First, to provide the reliable predictive models needed for control, a data-driven approach is introduced that learns computationally efficient models from simulation or experimental measurements. Unlike prevailing state-of-the-art methods that struggle to remain predictive and often violate core aerodynamics, this framework explicitly enforces physical principles, guaranteeing models that remain accurate and stable over long time periods. Second, a computational framework is developed to identify optimal finite-amplitude perturbations that produce the largest impact on the airflow. This isolates the system's inherent sensitivities, enabling the design of targeted control strategies capable of manipulating unsteady flows. Third, to obtain a small set of states for modeling that are physically meaningful and descriptive, a spectrally structured coordinate extraction method is developed that separates persistent wake harmonics from transient dynamics without sacrificing the spatial orthogonality required for consistent energy accounting. This yields a reduced set of modeling coordinates which describe the underlying physical mechanisms driving instabilities. Finally, building on these coordinates, the Block-Rotational Embedded Modulator Form (BREMF) provides a modeling architecture that exploits oscillatory structure in the coordinates to explicitly preserve lossless quadratic dynamics by construction. This method builds on the prior modeling framework and yields significant computational advantages while ensuring physically accurate models. Together, these contributions move beyond empirical fits of fluid data toward modeling and controlling the mechanics of the underlying airflow system. By linking bounded physics-informed regression, finite-amplitude sensitivity analysis, and spectrally organized coordinates, this dissertation provides a practical framework for the reduced-order modeling and control of complex airflows.

This dissertation develops physics-informed and data-driven methods for modeling, understanding, and controlling nonlinear aerodynamic flows. The work focuses on building reduced-order models that remain physically meaningful over long times, identifying sparse perturbations that strongly influence nonlinear flow evolution, and using time-frequency/phase-based analysis to understand separated-flow response to actuation.

When and Where: Timing and Placement of Pulses for Control of a 2D Separated Flow Over a Cambered Airfoil

Sergio A. Castiblanco-Ballesteros, A. Leonid Heide, Maziar S. Hemati, Gustaaf B. Jacobs

AIAA SCITECH 2026 Forum Conference

Paper

A parametric study of the sensitivity of location and timing of pulsed control on the aerodynamics of a NACA-65-412 airfoil at Re = 2 x 10^4 is presented. A net-zero-mass Dirac delta pulse in time with spatial Gaussian distribution is independently placed at six locations between the leading edge and the fixed separation location at the zero, averaged skin friction point. The pulse is also independently applied at six time instances within one period of the baseline flow's limit cycle. Lagrangian flow field analysis shows that this control triggers Kelvin-Helmholtz instabilities in the separated shear layer. A continuous wavelet analysis of lift and velocity signals reveals that sub-cycle variations in pulse timing, at fixed spatial positions, can flip the flow from pronounced lift suppression to transient reattachment, achieving up to 30% lift enhancement. A time-frequency analysis shows that vortex interlocking between instability-driven vortices and naturally occurring counter-rotating structures near the trailing edge underpins these dramatic transitions.

The Problem: Flow separation severely degrades aerodynamic performance by reducing lift and increasing drag. Modern pulse-jet actuators can alter these separated flows, but while finding the best physical location for an actuator has been widely studied, knowing the exact timing to fire the pulse within a natural vortex-shedding cycle remains largely unexplored.
The Goal: To map out how the sub-cycle timing and physical location of a pulsed jet independently and collectively influence separation dynamics and transient lift over a cambered airfoil. In other words, we wish to perform a parametric study to identify where and when the airfoil system is most sensitive to actuation.
The Outcome: Using continuous wavelet and Lagrangian flow analyses, we discovered that small temporal adjustments in pulse timing can radically flip the flow from lift reduction to transient reattachment. Leveraging this timing-dependent vortex interlocking mechanism allowed us to identify a timing and placement for actuation that gave up to a 30% enhancement in lift.

2025

A Convex Optimization Approach to Compute Trapping Regions for Lossless Quadratic Systems

Shih-Chi Liao, A. Leonid Heide, Maziar S. Hemati, Peter J. Seiler

International Journal of Robust and Nonlinear Control Journal IJ Robust Nonlinear Control

Paper

Quadratic systems with lossless quadratic terms arise in many applications, including models of atmosphere and incompressible fluid flows. Such systems have a trapping region if all trajectories eventually converge to and stay within a bounded set. Conditions for the existence and characterization of trapping regions have been established in prior work for boundedness analysis. However, prior solutions have used non-convex optimization methods, resulting in conservative estimates. In this paper, we build on this prior work and provide a convex semidefinite programming condition for the existence of a trapping region. The condition allows for precise verification or falsification of the existence of a trapping region. If a trapping region exists, then we provide a second semidefinite program to compute the least conservative radius of the spherical trapping region. Two low-dimensional systems are provided as examples to illustrate the results. A third high-dimensional example is also included to demonstrate that the computation required for the analysis can be scaled to systems of up to approximately O(100) states. The proposed method provides a precise and computationally efficient numerical approach for computing trapping regions. We anticipate this work will benefit future studies on modeling and control of lossless quadratic dynamical systems.

The Problem: Many complex systems in atmospheric science and fluid dynamics feature “lossless” quadratic interactions. While these systems are stable if they possess a mathematical “trapping region” (a boundary that all forecasts eventually converge inside), previous methods to find these regions relied on non-convex optimization, which produced overly conservative boundary estimates.
The Goal: To develop a convex optimization approach to mathematically prove the existence of these trapping regions and calculate the tightest possible physical boundaries for the system.
The Outcome: We introduced a convex semidefinite programming (SDP) framework that certifies whether a trapping region exists. If it does, our second optimization step calculates the tightest, least-conservative bounding radius possible.

An optimization framework for analyzing nonlinear stability due to sparse finite-amplitude perturbations

A. Leonid Heide, Maziar S. Hemati

Physical Review Fluids Journal Physical Review Fluids

Paper

Recent works have established the utility of sparsity-promoting norms for extracting spatially-localized instability mechanisms in fluid flows, with possible implications for flow control. However, these prior works have focused on linear dynamics of infinitesimal perturbations about a given baseflow. In this paper, we propose an optimization framework for computing sparse finite-amplitude perturbations that maximize transient growth in nonlinear systems. A variational approach is used to derive the first-order necessary conditions for optimality, which form the basis of our iterative direct-adjoint looping numerical solution algorithm. When applied to a reduced-order model of a sinusoidal shear flow at Re = 20, our framework demonstrates that energy injection into a single vortical mode yields comparable energy amplification to the non-sparse optimal solution, which concentrates 92% of the energy in the same mode. Subsequent analysis of the dynamic response of the flow establishes that these sparse optimal perturbations trigger many of the same nonlinear modal interactions that give rise to transient growth when all modes are perturbed in an optimal manner. It is also observed that as perturbation amplitude is increased, the maximum transient growth is achieved at an earlier time. Our results highlight the power of the proposed optimization framework for revealing sparse perturbation mechanisms for transient growth and instability in fluid flows. We anticipate the approach will be a useful tool in guiding the design of flow control strategies in the future.

The Problem: Traditional stability analysis can find the “worst-case” disturbances that trigger instability growth in fluid flows by identifying the sensitivities inherent to the system. However, these solutions typically require injecting energy across every single state or physical location of the system simultaneously. In real-world engineering, perturbing a whole system is rarely practical or possible, making this impractical for flow control.
The Goal: To find out where we can “kick” a nonlinear dynamical system to get the “biggest bang for our buck”. In other words, given a limited energy budget, can we get a large response in the system by targeting only a small number of locations or modes rather than distributing that energy throughout? The goal is then to find a (sparse) subset of states where we can apply a finite-amplitude perturbation such that we use minimal input energy to get the largest possible response.
The Outcome: Tested on a fluid shear flow model, our framework identified that kicking a small subset (or even just one vortical mode) yielded a similar energy amplification as a fully distributed perturbation. By isolating these highly sensitive single points of failure, the framework provides a roadmap for designing highly efficient, targeted flow-control strategies.

2022

A low-order nonlinear model of a stalled airfoil from data: Exploiting sparse regression with physical constraints

A. Leonid Heide, Katherine J. Asztalos, Scott T. M. Dawson, Maziar S. Hemati

AIAA AVIATION 2022 Forum Conference AIAA AVIATION

Paper

This work uses data-driven sparsity-promoting methods to obtain low-order governing equations for the wake of a stalled airfoil. Direct numerical simulation data of a NACA-0009 airfoil at an angle of attack of 15 degrees is utilized in this study, with actuation being performed by injecting momentum into the flow near the airfoil's leading edge. Proper Orthogonal Decomposition (POD) is used to obtain a reduced order representation of the flow field. The Sparse Identification of Nonlinear Dynamics (SINDy) framework is then implemented to obtain low-order quadratic governing equations for the flow over the stalled airfoil. The SINDy model is constrained to preserve the energy-conserving property of the quadratic nonlinearity and associated triadic energy-transfer mechanisms. Low-order nonlinear models of the unsteady flow field associated with the stalled airfoil are obtained and cross-validated using off-design data. Furthermore, an output equation that predicts the lift coefficient is also identified and cross-validated. These low-order nonlinear models are expected to facilitate future developments in model-based analysis and control of separated flows.

The Problem: Data-driven modeling methods (like Sparse Identification of Nonlinear Dynamics, or SINDy) can identify simplified equations for fluid flows, but they typically only track certain abstract flow states rather than engineering metrics of interest, such as aerodynamic lift. Furthermore, without physical constraints, standard data-fitting algorithms tend to violate basic energy laws over long periods, leading to fragile models that fail when tested outside of their narrow training conditions.
The Goal: To pair a physics-constrained, energy-preserving model of a stalled airfoil’s wake with a dedicated nonlinear output equation. In other words, the goal is to build a low-order model that tracks the physics of the flow field accurately over long periods, while also mapping those states directly to a predictive equation for the aerodynamic lift coefficient.
The Outcome: We successfully identified and cross-validated a compact, coupled model that accurately predicts lift under off-design and transient conditions. By forcing the underlying flow model to strictly obey physical energy conservation and directly embedding the lift calculation as a matched output equation, the resulting framework predicts aerodynamic forces with significantly greater accuracy and robustness than standard, unconstrained approaches.

2019

Evaluating the Accuracy of Conventional Drag Prediction and Validation Methods for Small Electric UAVs

Leonid Heide

University of Minnesota Undergraduate Research Opportunities Program Report UROP

Paper

The cost-effectiveness and versatility of unmanned aerial vehicles has led to increasing demand for small, fixed-wing, electric aircraft. Aircraft performance characteristics such as flight time, efficiency, and turn performance depend strongly on aerodynamic drag, making accurate drag prediction important for aircraft design, control, and mission planning. This report evaluates the accuracy of conventional component drag build-up methods for small electric UAVs and examines the use of glide tests for drag validation. Glide tests were performed on three aircraft from the University of Minnesota UAV lab, and wind-tunnel tests were conducted to measure drag generated by windmilling propellers. The component drag build-up method was then modified to account for small-UAV operating conditions and compared against measured drag data. The results show that, with appropriate corrections for effects such as turbulent boundary layers, bluff-body drag, windmilling propellers, and interference drag, component drag build-up methods can be adapted to predict drag for small electric UAVs. The report also demonstrates that glide tests are an effective low-cost method for measuring drag and validating predicted values.

The Problem: This work builds on the prior 2018 study (which proved that traditional methods underpredict small UAV drag by up to 71%). While high-fidelity Computational Fluid Dynamics (CFD) can simulate these low Reynolds number flight environments, CFD is expensive, time-consuming, and computationally restrictive for most drone developers, leaving them without a simple way to predict performance.
The Goal: To take the conventional Component Drag Build-Up Method (CDBM) and modify it to account for the unique physics of small UAVs, providing developers with a more accurate analytical tool that can be validated via simple, low-cost flight testing.
The Outcome: By analyzing three distinct drone geometries, we built a modified CDBM+ flight testing framework. The models achieve accuracy by modeling the UAV boundary layer as fully turbulent, utilizing Hoerner form-factors, adding mathematical penalties for un-filleted bluff noses, and incorporating wind-tunnel data to account for the resistance of windmilling propellers. Unpowered glide tests proved effective as a validation tool, with the newly predicted drag curves falling within the experimental 95% error bounds of the flight data.

2018

Evaluating the Impact of Parasitic Drag on the Accuracy of Endurance Predictions

Leonid Heide, Chris Regan, Demoz Gebre-Egziabher

University of Minnesota Undergraduate Research Opportunities Program Report UROP

Paper

The cost-effectiveness and versatility of electric unmanned aerial vehicles has led to increasing demand for efficient payload-carrying aircraft. Many design criteria require endurance predictions based on a specific mission profile, but these predictions rely strongly on accurate estimates of parasitic drag. This report evaluates how conventional component drag build-up methods affect endurance predictions for small electric UAVs. Glide tests were performed on an E-flite Ultra-Stick 25e equipped with an airspeed sensor, GPS, and flight controller, allowing drag to be estimated from measured descent rates and airspeeds. The experimentally inferred parasitic drag curve was then compared with predictions from a conventional component drag build-up method. The results showed that parasitic drag for the small UAV was substantially larger than predicted, producing large errors in drag and power estimates and demonstrating that conventional aircraft drag assumptions can lead to unreliable endurance predictions for small electric UAVs.

The Problem: When calculating how long a battery-powered, fixed-wing UAV can stay in the air, engineers rely on mathematical models to estimate aerodynamic drag. The standard approach is to employ the Component Drag Build-Up Method (CDBM), which uses empirical formulas designed for full-scale aircraft operating at high Reynolds numbers. Because small electric UAVs are tiny and fly slowly, they operate at much lower Reynolds numbers where viscous forces dominate, meaning assumptions about a “clean” body and low component interference fall apart.
The Goal: Experimentally test a small electric UAV using unpowered glide tests to back out its true parasitic drag curve, and determine if standard conventional drag estimation formulas are accurate enough to make valid flight endurance predictions.
The Outcome: Tested on an Eflite Ultra-Stick 25e model aircraft, the study revealed a gap: the standard CDBM underpredicted actual parasitic drag by an average error of 71%. Because the power required to overcome drag scales cubically with velocity, this underestimation means that conventional aircraft formulas overestimate battery life, and must be used cautiously when obtaining endurance predictions for small UAVs.