karhunen-loève expansion

This work addresses the challenge of high-fidelity reconstruction of physical fields under suboptimal observational conditions—such as noise, incomplete spatial coverage, or limited resolution—by introducing LatentPDE, a novel framework that explicitly parameterizes the latent variables of a diffusion model as the coefficients and source terms of governing partial differential equations (PDEs). This approach constructs a physically interpretable latent space by integrating structured sparsity modeling with PDE priors, enabling simultaneous field reconstruction and super-resolution at arbitrary resolutions. Moreover, it incorporates a mechanism for uncertainty quantification. Experimental results demonstrate that LatentPDE significantly outperforms existing physics-informed diffusion methods across diverse missing-data scenarios, achieving both high reconstruction accuracy and reliable uncertainty estimates.

This paper addresses the modeling challenge of deterministic inputs yielding strongly non-Gaussian stochastic outputs in neural systems. Methodologically, it introduces the first probabilistic neural architecture generation framework based on manifold-valued hidden random fields: neural topology and synaptic weights are jointly modeled as anisotropic Gaussian random fields on compact, boundaryless, multiply connected manifolds, with connectivity defined via geodesic distance and field affinity; geometrically aware sparsification is achieved through percentile-based diffusion masking; and scalable stochastic inference is enabled via inhomogeneous Poisson sampling coupled with Monte Carlo likelihood estimation. Theoretical contributions include establishing expressivity and well-posedness foundations for stochastic mappings, proving measurability, expressive variability, and feasibility of single-sample supervised learning. The framework requires no predefined network architecture and inherently integrates geometric awareness with statistical interpretability.

To address data sparsity, indirectness, and limited sample size in random field modeling for scientific and engineering applications, this paper proposes a domain-knowledge-integrated constrained generative framework. Methodologically, we design a physics- and statistics-constrained variational autoencoder (cVAE), pioneering the incorporation of hard constraints into latent-space learning; develop a functional decoder that inherently enforces prior physical or statistical laws in reconstructions; and integrate latent-space flow matching for efficient, controllable probabilistic modeling. Our key contributions are: (i) the first synergistic optimization of physical constraints and flow matching within the VAE latent space; and (ii) state-of-the-art performance on wind velocity field reconstruction and material property inversion—achieving high-fidelity random field generation and rigorous uncertainty quantification from only a few indirect observations, significantly outperforming unconstrained baselines.

This work addresses the limitations of classical perturbation analyses for CUR decomposition, which rely solely on global noise levels and fail to capture how sampling structures influence local reconstruction errors. By employing a local tangent space expansion, the study establishes, for the first time, a precise connection between the first-order perturbation error of rank-truncated CUR mappings with fixed index sets and the oblique projection operators induced by sampling. This reveals a mechanism whereby invisible perturbations are automatically eliminated under first-order approximation. Leveraging Fréchet derivatives and local Taylor expansions, the authors theoretically derive first- and second-order local convergence rates. Numerical experiments further validate the perturbation-removal effect across distinct subspaces, highlighting CUR’s unique advantage over truncated SVD in structural sensitivity.

This paper addresses unsupervised representation learning for sequential data. We propose a novel probabilistic flow decomposition framework that disentangles the latent-space dynamics into two orthogonal vector fields: a sparse curl-free field (corresponding to an irrotational potential field) and a divergence-free field (corresponding to a solenoidal rotational field), with sparsity priors newly imposed on both components. Within a variational autoencoder framework, our method jointly optimizes representation encoding, velocity field estimation, and field-structure inference, implicitly learning approximately equivariant representations. Compared to prior approaches, our model simultaneously achieves static representation disentanglement and independence of dynamic transformation primitives, yielding significant improvements in data likelihood and unsupervised equivariance error across multiple sequence transformation benchmarks—achieving state-of-the-art performance. Crucially, the learned vector fields admit clear physical interpretations grounded in classical vector calculus.

This work proposes a novel tensor decomposition framework, termed HMD, which addresses a key limitation of traditional low-rank methods such as Tucker and CP decompositions: their inability to capture high-order interactions among modes, as they only model mode-wise independent variations. HMD explicitly incorporates inter-modal high-order couplings within the low-rank approximation by introducing specially designed projection operators that jointly encode both isolated mode-specific effects and cross-modal interaction structures. By transcending the structural constraints inherent in conventional decompositions, the proposed method achieves substantially lower reconstruction errors and demonstrates superior fidelity and robustness across three diverse benchmark datasets, consistently outperforming Tucker and CP decompositions.

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 work addresses the limitations of fixed-complexity generative priors in inverse problems, which often suffer from insufficient expressiveness or overfitting to noise. The authors propose an adaptive-complexity generative prior that dynamically modulates the capacity of diffusion models, normalizing flows, and variational autoencoders via a nested dropout mechanism, tailoring model complexity to the specific demands of each inverse problem. This approach is the first to enable continuous and controllable complexity adjustment across multiple generative model families, accompanied by theoretical guarantees in linear denoising settings. Experimental results demonstrate significant improvements over fixed-complexity baselines across diverse tasks—including compressive sensing, image inpainting, denoising, and phase retrieval—with markedly reduced reconstruction errors.

This study addresses accuracy and consistency issues in the numerical construction of the Karhunen–Loève expansion (KLE) arising from discretization, quadrature rules, and finite sample sizes. It establishes an algebraic equivalence between the spectral decomposition of the Fredholm integral equation and the singular value decomposition (SVD) of a weighted sample covariance matrix, thereby unifying model-driven and data-driven KLE frameworks. The work innovatively constructs the covariance function on a non-simply-connected three-dimensional toroidal domain using the shortest interior path distance, and implements the approach numerically with unstructured meshes and Gaussian quadrature. Experiments demonstrate that, in a one-dimensional benchmark problem, SVD-based eigenvalue estimates and empirical KL coefficients converge to the theoretical 𝒩(0,1) distribution. In two-dimensional irregular and three-dimensional toroidal domains, the study systematically quantifies the combined influence of discretization strategy, quadrature accuracy, and sample size on KLE reconstruction error.

Traditional generative models rely on dense grid representations, which struggle to accurately reconstruct sharp transient features in physical signals. This work proposes a generative diffusion framework based on singularities in the complex plane, achieving high-fidelity signal reconstruction from degraded or partial observations by learning singularity configurations that satisfy physical constraints. The approach introduces a structurally stable, resolution-independent, and interpretable singularity representation that effectively mitigates Gibbs artifacts and enables reconstruction on arbitrary grids. A Transformer-based diffusion model directly predicts complex-plane singularity coordinates consistent with geometric constraints. In Burgers' shock-wave tasks, the method surpasses the reconstruction accuracy of a 1024-point grid using only 32 singularities, reduces zero-shot sub-resolution generalization error by 4.2×, and achieves a physical parameter recovery error of 10⁻⁴.