multiscale pod

This work addresses the limitations of traditional Proper Orthogonal Decomposition (POD) in convection-dominated and turbulent systems, where slow Kolmogorov n-width decay necessitates a large number of modes and often overlooks critical low-energy structures. The authors propose a novel framework that combines linear POD encoding with nonlinear neural network decoding, uniquely integrating LassoNet’s hierarchical sparsity mechanism into a joint optimization of mode selection and manifold learning. This approach simultaneously minimizes reconstruction error and automatically identifies the most informative modes, balancing representational capacity, accuracy, and physical interpretability. In benchmark tests involving convection-dominated and chaotic flows, the method matches or exceeds state-of-the-art performance; notably, for turbulent channel flow at $Re_\tau = 5200$, it reduces reconstruction error by 51%–78% compared to polynomial manifold-based methods.

This work proposes Neural-POD, a novel framework that overcomes the limitations of traditional AI-for-Science approaches, which often fail to generalize across new parameters or discretizations due to dependence on fixed grids or resolutions. By constructing nonlinear orthogonal bases in infinite-dimensional function spaces via neural networks, Neural-POD reformulates basis construction as a sequence of residual minimization problems, analogous to a nonlinear, learnable Gram–Schmidt process that incrementally captures data structure. The method transcends the linearity constraints of classical Proper Orthogonal Decomposition (POD), enabling optimization under arbitrary norms, resolution-invariant mappings, and effective nonlinear feature extraction. It is designed for seamless integration into reduced-order modeling and operator learning pipelines. Numerical experiments on complex spatiotemporal systems—including the Burgers and Navier–Stokes equations—demonstrate its robustness and efficacy in bridging classical model reduction with modern operator learning paradigms.

This study addresses the challenges of high-fidelity Proper Orthogonal Decomposition (POD), which is prone to overfitting and computationally expensive when snapshot data are scarce. To mitigate these issues, the authors propose a multi-fidelity POD framework that integrates low-fidelity model data via a controlled-variable approach, enabling weighted coupling of high- and low-fidelity snapshots. This integration yields an unbiased estimator of the projection error, which is then leveraged to optimize the POD subspace. The method substantially reduces the variance of the error estimator under limited computational budgets, enhancing subspace robustness and effectively alleviating overfitting. Demonstrated on Pine Island Glacier velocity modeling, the approach achieves accuracy comparable to single-fidelity POD while requiring only one-tenth of the offline snapshot generation cost.

This study investigates how to extract scale-selective dominant modes from Transformer attention fields and quantify inter-layer complexity. To this end, it introduces Proper Orthogonal Decomposition (POD)—a technique from turbulence analysis—into the study of attention mechanisms for the first time, combining it with the Morlet continuous wavelet transform to construct a multi-scale analysis framework that requires neither architectural modifications nor linguistic annotations. The work further proposes a spectral concentration index to measure the complexity of individual layers. Experimental results reveal a systematic scale organization within Transformers: shallow layers emphasize fine-grained patterns, while deeper layers shift toward coarse-grained structures. Additionally, the framework provides data-driven effective rank estimates for each layer, significantly enhancing the understanding of the intrinsic structure of attention mechanisms.

In parametric dynamical systems, the Proper Orthogonal Decomposition (POD) basis drifts with parameters, degrading the accuracy of reduced-order models (ROMs). Method: This paper proposes the Projected Gaussian Process (pGP) framework—the first to formulate subspace adaptation as a statistical learning task mapping parameter space to the Grassmann manifold. It employs a two-stage geometric mapping: Euclidean space → horizontal space → Grassmann manifold, integrating POD, exponential/logarithmic maps, horizontal-space projection, and Gaussian process regression to enable uncertainty-aware POD subspace prediction while preserving manifold structure. Contribution/Results: Numerical experiments demonstrate that pGP significantly improves ROM accuracy and robustness in both parametric extrapolation and interpolation scenarios, and provides interpretable, calibrated confidence quantification—establishing a new paradigm for parameter-sensitive model reduction.

This work addresses the computational expense and difficulty in uncertainty quantification associated with super-resolving high-dimensional spatial fields when modeling directly in pixel space. The authors propose PODiff, the first method to introduce diffusion models into the coefficient space of Proper Orthogonal Decomposition (POD), constructing a conditional generative framework within a fixed, variance-ordered latent space. By leveraging the orthogonality of POD modes, PODiff establishes an interpretable latent geometry that enables efficient and structure-preserving ensemble generation. Evaluated on sea surface temperature downscaling and convection–diffusion benchmark tasks, PODiff achieves reconstruction accuracy comparable to pixel-space diffusion models with substantially lower memory consumption and demonstrates superior uncertainty calibration compared to both deterministic approaches and Monte Carlo Dropout.

This work addresses the challenges of efficiently sampling posterior distributions in Bayesian inversion when confronted with high-dimensional parameter spaces, sparse data, and strong noise, which hinder conventional dimensionality reduction techniques. The authors propose the α-likelihood informed subspace (α-LIS) method, which rigorously extends likelihood-informed subspace (LIS) theory to tempered posteriors with α ∈ [0,1], enabling the construction of a partially informed low-dimensional subspace for effective dimension reduction. By integrating data from multiple tempering levels and incorporating a gradient-free approximation strategy, the approach significantly enhances robustness and sampling efficiency in scenarios where gradients are unavailable or observations are highly noisy. Both theoretical analysis and numerical experiments demonstrate that near-optimal dimension reduction can be achieved with relatively small α values, yielding overall performance superior to traditional methods restricted to α = 1.

High-fidelity simulations, such as computational fluid dynamics and finite element analysis, are essential for modeling complex engineering systems but are often prohibitively expensive for tasks including parametric studies, optimization, and real-time control. Projection-based reduced-order models (ROMs) alleviate this cost by projecting the governing dynamics onto low-dimensional subspaces. However, their performance can deteriorate under parameter variation, motivating the need for adaptive basis construction. In this work, we propose a constrained ensemble learning framework, termed Constrained Extreme Gradient Boosting (cXGBoost), for predicting Proper Orthogonal Decomposition (POD) bases as functions of system parameters. The approach leverages a geometric representation of subspaces on the Grassmann manifold, which are mapped to a Euclidean space to enable efficient regression using gradient boosting trees. A norm constraint is imposed during training to ensure the validity of the inverse mapping and preserve the geometric structure of the predicted subspaces. The proposed method is evaluated on four numerical examples, including fluid dynamics and wave propagation problems, demonstrating its ability to accurately predict parameter-dependent bases while maintaining robustness across nonlinear regimes. These results highlight the potential of combining geometric learning with constrained ensemble methods for scalable and reliable reduced-order modeling of high-dimensional parametric systems.

Traditional linear model reduction techniques, such as Proper Orthogonal Decomposition (POD), struggle to effectively handle high-dimensional dynamical systems featuring sharp gradients. This work proposes the GNN-LaSDI framework, which integrates a graph autoencoder for nonlinear dimensionality reduction and employs operator learning to directly model temporal evolution in the latent space. The approach accurately captures the locations of steep gradients while maintaining computational efficiency. Furthermore, it introduces a novel point-cloud–oriented error metric that provides a more intuitive assessment of local accuracy. In two representative numerical experiments, GNN-LaSDI achieves significantly higher accuracy than POD-LaSDI and substantially reduces computational cost compared to GD-LSPG, thereby striking an effective balance between accuracy and efficiency.

This work addresses the challenge of efficiently and accurately solving multiscale elliptic partial differential equations with strongly heterogeneous or highly oscillatory coefficients, where conventional numerical methods suffer from high computational cost and existing neural operators lack sufficient accuracy. The authors propose LOD-MSNO, a hybrid model that uniquely integrates problem-adapted basis functions from the Local Orthogonal Decomposition (LOD) method as a strong prior within a neural operator framework. By leveraging data-driven learning, the approach alleviates the computational bottlenecks inherent in LOD while providing theoretical error estimates for coefficient learning. Numerical experiments demonstrate that, under strongly heterogeneous multiscale inputs, LOD-MSNO significantly outperforms current neural operator baselines, achieving markedly higher accuracy without compromising computational efficiency.