LSDO Lab
Methods

Graph-native modeling for automated adjoint sensitivity analysis

Context
Gradient-based multidisciplinary design requires total derivatives through explicit calculations, nonlinear equation solves, and time-dependent models.
Gap
Implementing adjoints and coordinating derivative calculations across these models can require substantial application-specific code.
Research goal
Generate model evaluations and adjoint sensitivities from a common computational graph, including coupled PDE and ODE models.

Selected studies

References
Graph-based model compilationDifferentiable time integrationPDE model composition

Generating model outputs and sensitivities from one description

CSDL records model equations as a graph, transforms that graph, and generates executable evaluations and adjoint sensitivities. Two engineering examples require roughly half the user-written code of their original implementations, with no measurable increase in computation time.

Primary paper

Victor Gandarillas, Anugrah Jo Joshy, Mark Z. Sperry, Alexander K. Ivanov, John T. Hwang. A Graph-Based Methodology for Constructing Computational Models that Automates Adjoint-Based Sensitivity Analysis. Structural and Multidisciplinary Optimization 67.5 (2024), p. 76, 2023.

PDF
Source code is transformed into a computational graph and then into executable code that returns model outputs and total derivatives.
CSDL transforms source code into a computational graph, applies graph transformations, and generates executable model evaluations and total derivatives. Fig. 3, Gandarillas et al., 2024 [1]
Stacked bars compare the fractions of sparse and dense operation-partial Jacobians across engineering models.
Sparse and dense operation-partial derivatives vary substantially across demonstrated MDO models. Fig. 7, Sperry et al., 2023 [2]
Gantt charts show operation schedules and communication for an eight-rotor model executed on one, four, and eight processors.
Static scheduling assigns eight-rotor blade-element-momentum operations and their communication to one, four, and eight processors. Fig. 8, Sperry et al., 2024 [3]
Detailed computational graph of a VortexAD linear-potential-flow analysis, showing dependencies among aerodynamic operations.
VortexAD expresses an unsteady linear-potential-flow solver as a computational graph, exposing the dependencies needed for automatic differentiation. Fig. 1, Scotzniovsky and Hwang, 2026 [4]

Embedding time-dependent dynamics in gradient-based optimization

Ozone computes sensitivities through an ODE solution and the surrounding design model. A common interface supports explicit and implicit Runge–Kutta methods with time-marching, checkpointing, and parallel-in-time approaches, allowing their convergence, runtime, and memory costs to be compared on the same optimization problem.

Primary paper

Mark Z. Sperry, John T. Hwang. Ozone: an open-source ordinary differential equation solver for gradient-based optimization. Optimization and Engineering, 2025.

DOIPDF
Design-structure matrix for a time-dependent model implemented with Ozone and CSDL.
Ozone connects an ODE integrator to upstream and downstream CSDL models in a larger optimization problem. Fig. 6, Sperry and Hwang, 2025 [5]
Performance comparison for trajectory-optimization cases solved using Ozone.
Trajectory-optimization comparisons show convergence, derivative-verification, time, and memory behavior across Ozone solution approaches. Fig. 12, Sperry and Hwang, 2025 [5]

Automating adjoint sensitivities for PDE-based multidisciplinary models

feMo uses FEniCSx to differentiate finite-element residuals and CSDL to combine those derivatives with other disciplines through the adjoint method. Verification problems, electric-motor shape optimization, and an eVTOL wing gust-response analysis demonstrate the coupling of PDE and non-PDE models.

Primary paper

Ru Xiang, Sebastiaan P. C. van Schie, Luca Scotzniovsky, Jiayao Yan, David Kamensky, John T. Hwang. Automating adjoint sensitivity analysis for multidisciplinary models involving partial differential equations. Structural and Multidisciplinary Optimization (in press 2024, per CV), 2024.

DOIPDF
feMo couples a CSDL interface for inputs, states, and outputs to a finite-element-analysis backend with PDE residuals, boundary conditions, solvers, and derivatives.
feMo connects PDE-based finite-element models to CSDL so coupled multidisciplinary models can use a shared adjoint-sensitivity workflow. Fig. 3, Xiang et al., 2024 [6]

References

  1. Victor Gandarillas, Anugrah Jo Joshy, Mark Z. Sperry, Alexander K. Ivanov, John T. Hwang. A Graph-Based Methodology for Constructing Computational Models that Automates Adjoint-Based Sensitivity Analysis. Structural and Multidisciplinary Optimization 67.5 (2024), p. 76, 2023.
    PDF
  2. Mark Sperry, Kavish Kondap, John T. Hwang. Automatic adjoint sensitivity analysis of models for large-scale multidisciplinary design optimization. AIAA AVIATION 2023 Forum, 2023.
    DOIPDF
  3. Mark Sperry, Kavish Kondap, John T. Hwang. Static Task Scheduling for Parallel Execution of Large-Scale Multidisciplinary Design Optimization Models. AIAA SCITECH 2024 Forum, 2024.
    DOIPDF
  4. Luca Scotzniovsky, John T. Hwang. VortexAD: A Graph-Based Linear Potential Flow Analysis Framework for Large-Scale MDO. AIAA AVIATION 2026 Forum, AIAA 2026-4642, 2026.
    PDF
  5. Mark Z. Sperry, John T. Hwang. Ozone: an open-source ordinary differential equation solver for gradient-based optimization. Optimization and Engineering, 2025.
    DOIPDF
  6. Ru Xiang, Sebastiaan P. C. van Schie, Luca Scotzniovsky, Jiayao Yan, David Kamensky, John T. Hwang. Automating adjoint sensitivity analysis for multidisciplinary models involving partial differential equations. Structural and Multidisciplinary Optimization (in press 2024, per CV), 2024.
    DOIPDF

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