adjoint reduced-order modeling

This work addresses the challenge of efficiently and accurately estimating quantities of interest (QoI) in multi-query linear problems, where conventional approaches suffer from high computational costs and strong dependence on load configurations. The authors propose a novel reduced-order modeling paradigm based on the adjoint problem, shifting the focus of model reduction from the primal to the adjoint equation for the first time. By introducing a parameterized kernel function to replace the full external load, the method constructs a load-independent surrogate model. Demonstrated on Poisson’s equation and plane-stress elasticity problems, the approach achieves rapid convergence and significantly outperforms traditional primal-based reduction strategies. It enables high-fidelity QoI estimation while supporting fast multi-scenario evaluation and virtual chart generation, thereby greatly enhancing the generality and efficiency of early-stage design optimization.

This work proposes a novel framework that integrates continuous-time operator inference with the adjoint-state method to address the poor accuracy and unstable extrapolation of traditional data-driven reduced-order models under sparse sampling and noisy data. By minimizing trajectory loss during training, the approach avoids direct differentiation of noisy measurements and leverages temporal integration for intrinsic regularization. For the first time, the adjoint method is incorporated into continuous-time operator inference, enabling efficient gradient computation and stable optimization. Combining continuous adjoint equations, projected snapshot matching, and gradient-based optimization, the method demonstrates significantly improved accuracy and rolling prediction stability over standard operator inference when tested on the Burgers, Fisher–KPP, and convection–diffusion equations under sparse or noisy data conditions.

This work addresses the high computational cost of gradient-based optimization for transient convection–diffusion problems using reduced-order models (ROMs). We propose a “optimize-then-reduce” coupled framework wherein the primal and adjoint systems are simultaneously solved within the reduced space at each time step. To alleviate the dependency on high-dimensional adjoint snapshots, we devise an efficient adjoint snapshot collection strategy. Furthermore, we overcome conventional limitations in adjoint basis construction by introducing an adaptive adjoint basis selection method guided by energy decay, iteration count, and computational time. Numerical experiments demonstrate that the proposed approach significantly reduces the computational overhead of adjoint system solves and overall simulation time, while maintaining controllable error levels. The method enables real-time, high-fidelity optimization in ROM–ROM coupled settings.

To address the high computational cost of optimizing nonlinear responses in lightly damped mechanical systems, this paper develops a dynamics optimization framework based on spectral submanifold (SSM)-based reduced-order models (ROMs). Methodologically, it innovatively integrates the adjoint method into SSM backbone curve sensitivity analysis, enabling efficient gradient computation for arbitrary-order polynomial parameters—a first in the literature. It further proposes an error-tolerance-driven adaptive ROM order selection strategy that balances accuracy and efficiency. The proposed approach significantly reduces computational overhead for high-dimensional parametric optimization while achieving precise customization of nonlinear frequency–amplitude response curves across multiple numerical benchmarks. This work advances SSM theory toward engineering practice and establishes a scalable, high-fidelity paradigm for nonlinear structural dynamics optimization.

For large-scale aerospace simulations—such as rotating detonation rocket engines—with state dimensions reaching tens of millions, conventional reduced-order modeling (ROM) becomes infeasible on a single machine. This work proposes distributed Operator Inference (dOpInf), the first framework enabling fully scalable, physics-constrained ROM construction. dOpInf integrates hybrid MPI/OpenMP parallelism, distributed linear algebra, proper orthogonal decomposition (POD) projection, and structured system identification. Deployed on high-performance computing platforms, it overcomes memory and computational bottlenecks inherent to monolithic ROM training, enabling highly concurrent ROM construction across thousands of CPU cores. Validated on a 2D channel flow problem, the resulting ROM preserves physical consistency while achieving extreme model compactness and a 100× speedup over full-order simulation. This efficiency facilitates computationally intensive engineering tasks, including design space exploration and uncertainty quantification.

This work proposes a hyper-reduction-free projected reduced-order Newton solver framework for polynomial nonlinear dynamical systems, addressing the computational overhead and complexity typically incurred by hyper-reduction in model order reduction. The method precomputes all projected residuals and Jacobian operators during the offline phase, achieving strict decoupling between offline and online stages. It is the first to enable fully hyper-reduction-free, efficient reduced-order modeling within both Galerkin and least-squares Petrov–Galerkin (LSPG) frameworks, while handling non-polynomial nonlinearities via lifting transformations. Numerical experiments on the Burgers equation and a heat equation with cubic reaction terms demonstrate that the proposed HRF-G and HRF-LSPG methods achieve speedups of approximately 100× and 10×, respectively, with state prediction errors below 10⁻², substantially reducing online costs while preserving high accuracy.

This work systematically compares neural operators and polynomial surrogates within a unified framework to reduce the computational cost of repeatedly solving parametric partial differential equations, with a focus on how surrogate performance depends on the regularity of input fields. Employing methods including Fourier Neural Operators, reduced-basis neural operators (trained with respect to $L^2_\mu$ and $H^1_\mu$ losses), reduced-basis sparse grids, and tensor-train polynomials, the study reveals for the first time that surrogate efficacy is highly sensitive to input smoothness: polynomial approaches exhibit markedly superior data efficiency when inputs are smooth ($s \geq 2$), whereas Fourier Neural Operators converge fastest for rough inputs ($s \leq 1$). Furthermore, incorporating derivative information during training significantly enhances accuracy and efficiency in low-data regimes.

This study addresses the computational challenge of efficiently simulating high-dimensional, nonlinear aeroelastic–flight dynamics coupled systems. The authors propose a general nonlinear model order reduction framework that constructs a second-order Taylor expansion of the residual around equilibrium points—sufficient to accurately represent cubic nonlinearities without requiring third-order terms—and employs a bi-orthogonal low-dimensional subspace spanned by the left and right eigenvectors of the Jacobian matrix to achieve an optimal projection. By integrating a matrix-free finite difference approximation, the method avoids dependence on the full-order model’s internal structure. Validated across three test cases of increasing complexity, the approach reduces system dimensionality from thousands to single digits, achieving speedups of up to 600× while accurately capturing strong nonlinear dynamic behaviors such as large deformations exceeding 10% of the wingspan.

This work addresses the limitations of traditional model-based reduced-order modeling in scenarios where high-fidelity model code is difficult to integrate, necessitating data-driven alternatives. It presents the first systematic integration of data-driven techniques—such as Dynamic Mode Decomposition—with model-driven approaches within the open-source library pyMOR. Built upon a unified interface of VectorArray, Operator, and Model abstractions, the proposed framework enables a flexible and efficient hierarchical reduction pipeline. The study demonstrates seamless interoperability between data-driven and model-driven methods, validates the efficacy of data-driven reduction through practical case studies, and highlights its performance advantages and complementary potential relative to conventional approaches.