Research

Our research is organized around three pillars: (1) structured representations of nonlinear dynamics, (2) operator-theoretic analysis and interpretable reduced coordinates, and (3) optimization, control, and learning of dynamical systems.

Research

1. Structured representations of nonlinear dynamics

A quasi-periodic trajectory on a torus—the torus time-spectral method solves for the solution directly on this manifold.

How do we build compact, structured representations of nonlinear time-dependent dynamics?

We develop spectral and frequency-domain frameworks that replace brute-force time marching with structure-exploiting representations of periodic and quasi-periodic behavior. Our work follows a natural progression: represent the motion via time-spectral methods, generalize to multi-frequency motion via torus methods, and characterize stability of the represented motion via Floquet theory.

Key contributions include the torus time-spectral method (TTSM) that lifts governing equations to an extended angular phase space with spectral convergence, spectral Floquet analysis for orbital stability of periodic systems, and time-spectral resolvent analysis for frequency response of periodically varying base flows.

Publication:

	Sicheng He, Hang Li, Kivanc Ekici. Torus Time-Spectral Method for Quasi-Periodic Problems arXiv preprint (2025).
	Sicheng He, Rohit Kanchi. Torus Time-Spectral Method for Three-Dimensional Wing Oscillations with Two Incommensurate Frequencies in preparation.
	Sicheng He, Max Howell, Dan Wilson. Spectral Floquet Analysis in preparation.
	Max Howell, Sicheng He. Time-Spectral Resolvent Analysis for Periodic Dynamical Systems arXiv preprint (2026).

2. Operator-theoretic analysis and interpretable reduced coordinates

Velocity resolvent response mode on the NASA Common Research Model wing, computed using our matrix-free resolvent analysis framework.

How do we extract the dominant mechanisms, coordinates, and sensitivities from large-scale nonlinear systems?

We build modal and operator-based tools to turn simulation data or linearized operators into understanding: what modes matter, what forcing/response structures dominate, and how sensitivities propagate through modal objects. Critically, our analysis tools are not passive diagnostics—they are made optimization-ready through differentiable formulations that connect directly to gradient-based design.

Key contributions include the first fully matrix-free resolvent analysis for 3D aerodynamic systems (NASA CRM, 1.8 million cells), differentiable resolvent analysis for flow control optimization, and differentiable POD for optimization-compatible modal decompositions and field inversion.

Publication:

	Sicheng He, Rohit Kanchi. Matrix-Free Resolvent Analysis for Large-Scale Aerodynamic Systems in preparation.
	Sicheng He, Shugo Kaneko, Max Howell, Daning Huang, Chi-An Yeh, Joaquim R. R. A. Martins. Large-Scale Flow Control Performance Optimization via Differentiable Resolvent Analysis in preparation.
	Rohit Sunil Kanchi, Sicheng He. Modal-Centric Field Inversion via Differentiable Proper Orthogonal Decomposition arXiv preprint (2026).

3. Optimization, control, and learning of dynamical systems

How do we control, optimize, and learn within structured dynamical representations?

Once dynamics are represented and interpreted, we ask: how do we modify them—suppress instability, improve performance, learn closures, and design systems with dynamics as first-class constraints? We combine adjoint methods, multidisciplinary optimization, and scientific machine learning to design, stabilize, and infer complex engineering systems governed by multiscale dynamics.

Key contributions include adjoint-based stability-constrained design optimization, Hopf-bifurcation instability suppression via the first Lyapunov coefficient, adjoint-based control co-design, a fundamental reverse algorithmic differentiation method for complex analytic functions yielding the first succinct eigenvalue derivative formula for general complex matrices, gradient-enhanced neural network surrogates for real-time aerodynamic analysis (Webfoil), and the differentiable Kalman filter for physics-informed state estimation with 90% error reduction.

Publication:

	Sicheng He, Eirikur Jonsson, Jichao Li, Joaquim R. R. A. Martins. Adjoint-Based Design Optimization of Stability Constrained Systems AIAA Journal (2024).
	Sicheng He, Max Howell, Daning Huang, Eirikur Jonsson, Galen W. Ng, Joaquim R. R. A. Martins. Adjoint-based Hopf-bifurcation Instability Suppression via First Lyapunov Coefficient arXiv preprint (2025).
	Sicheng He, Shugo Kaneko, Eirikur Jonsson, Marco Mangano, Joaquim R. R. A. Martins. Control co-design sensitivity computation using the adjoint method submitted to SIAM applied dynamics (SIADS) (2022).
	Sicheng He, Eirikur Jonsson, Joaquim R. R. A. Martins. Adjoint-based Limit Cycle Oscillation Instability Sensitivity and Suppression Nonlinear dynamics (2022).
	Sicheng He, Yayun Shi, Eirikur Jonsson, Joaquim R. R. A. Martins. Eigenvalue problem derivatives computation for a complex matrix using the adjoint method MSSP (accepted) (2023).
	Sicheng He, Eirikur Jonsson, and joaquim R. R. A. Martins. Derivatives for Eigenvalues and Eigenvectors via Analytic Reverse Algorithmic Differentiation AIAA Journal (2022).
	Jichao Li, Sicheng He, Mengqi Zhang, Joaquim R. R. A. Martins, Boo Cheong Khoo. Physics-Based Data-Driven Buffet-Onset Constraint for Aerodynamic Shape Optimization AIAA Journal (2022).
	Mohamed Amine Bouhlel, Sicheng He, and Joaquim R. R. A. Martins. Scalable gradient-enhanced artificial neural networks for airfoil shape design in the subsonic and transonic regimes Structural and Multidisciplinary Optimization (2020). (Webfoil)
	Jichao Li, Sicheng He, and Joaquim R. R. A. Martins. Data-driven constraint approach to ensure low-speed performance in transonic aerodynamic shape optimization Aerospace Science and Technology (2019).
	Yuan Wu, Sicheng He. DKFNet: Differentiable Kalman Filter for Field Inversion and Machine Learning arXiv preprint (2025).

Past research projects — aeroelastic optimization, wind turbine MDO, laminar-turbulent transition, structural global optimization.