a posteriori error estimation

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.

Neural operators (NOs) suffer from error accumulation and instability in long-term forecasting of nonlinear time-varying PDEs due to autoregressive inference. To address this, we propose ANCHOR—an online adaptive hybrid inference framework. Methodologically, ANCHOR integrates a pre-trained NO with physics-informed residual modeling and multi-scale PDE representation, while introducing two key innovations: (i) a ground-truth-free PDE residual estimator based on exponential moving average, enabling unsupervised online error monitoring; and (ii) the first incorporation of numerical analysis–inspired adaptive time stepping into the NO inference loop, dynamically triggering classical solver corrections. Evaluated on 1D/2D Burgers’, 2D Allen–Cahn, and 3D heat equations, ANCHOR improves long-term prediction stability by over 2× compared to standard NOs, while achieving inference speeds 2–3 orders of magnitude faster than high-accuracy numerical solvers.

Existing non-intrusive reduced-order modeling (ROM) methods suffer from limited interpretability and structural flexibility. To address this, we propose a regularized kernel interpolation framework grounded in reproducing kernel Hilbert spaces (RKHS). Our method explicitly embeds physics-informed feature maps and nonlinear closure terms into the ROM structure, enabling optimal low-dimensional dynamical approximation without requiring access to full-order model operators. This work is the first to integrate kernel interpolation with ROM structural constraints, thereby unifying model interpretability and representational flexibility. We further derive a computable a posteriori error bound that jointly accounts for projection error and kernel approximation error. Numerical experiments demonstrate that, compared to classical operator inference approaches, our method achieves higher predictive accuracy, superior generalization across diverse problems, and enhanced robustness—while preserving explicit correspondence between the ROM structure and the underlying physical system.

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.

In the performatively predictive setting, model deployment induces distributional shifts, hindering convergence to a performative stable point. To address this, we propose Affine Risk Minimization (ARM), an algorithm that explicitly models the predictive impact on data distributions by affinely combining historical model parameters—leveraging past training snapshots with learned weights. Theoretically, we establish the first unified convergence upper bound for both existing and new algorithms, proving its tightness; reveal that historical information can break the single-iteration lower-bound barrier; and introduce the first lower-bound analysis framework for ARM-style methods. Empirically, ARM achieves faster convergence and higher accuracy at the performative stable point across multiple benchmarks.

This work addresses the limited predictive diversity and poor out-of-distribution (OOD) uncertainty quantification of single pretrained models under distribution shift. The authors propose a Perturb-and-Correct approach that constructs a posterior ensemble using only a single model by applying random perturbations to hidden layers of the pretrained network and subsequently correcting them via least-squares affine transformations. This method uniquely exploits the affine redundancy inherent in neural networks, enhancing OOD prediction diversity and uncertainty calibration without compromising in-distribution performance. Empirical evaluations demonstrate that the proposed technique achieves a superior or competitive trade-off between in-distribution accuracy and OOD detection on MuJoCo dynamics prediction and CIFAR-10 OOD benchmarks compared to existing posterior ensemble baselines.

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 addresses the challenge of parameter optimization in computer model calibration, where the goal is to minimize discrepancies between multidimensional model outputs and observed data. The authors propose a novel root-finding paradigm that reformulates calibration as a root-search problem by constructing signed-residual-based Kriging or stochastic Kriging surrogates. Their approach integrates a sequential search space reduction strategy with a new acquisition function compatible with first-order optimizers. Notably, it guarantees algorithmic robustness even when roots may not exist—a scenario unaddressed by prior methods. Empirical evaluations demonstrate that the proposed method significantly outperforms conventional calibration techniques across both data-driven and physics-based modeling tasks, achieving higher computational efficiency while enhancing solution robustness.

This work addresses the closure errors and spurious spectral artifacts commonly arising in data-driven Koopman methods due to the lack of invariance in the learned feature space, which undermines prediction reliability. To overcome this, the authors propose a unified posterior diagnostic framework that quantifies invariance via principal angles and introduces Principal Angle Decomposition (PAD) of observables as a principled alternative to conventional SVD truncation. By integrating multi-step pointwise error bounds in a reproducing kernel Hilbert space (RKHS) with Gaussian process-based error surrogates, the framework enables certifiable dictionary learning and spectral analysis. Experiments on chaotic systems, high-dimensional benchmarks, and real-world datasets—including cavity flow and the Pluto–Charon system—demonstrate substantial improvements in modal accuracy and predictive reliability.