PDE problems with multi-scale features present significant challenges in scientific computing. While conventional deep learning methods based on multi-layer perceptrons exhibit spectral bias in resolving multi-scale features, physics-informed Kolmogorov–Arnold networks mitigate this issue through their architectural design and show advantages in representing multi-scale physical fields. On the other hand, insights from information bottleneck theory suggest that high-resolution sampling is important for capturing multi-scale behavior, but this requirement often leads to increased computational cost. To address this issue, we propose a simple yet effective multi-resolution training-enhanced Kolmogorov–Arnold network framework, which trains the model sequentially or alternately at different resolutions. The proposed framework is validated on various forward and inverse PDE problems with multi-scale features. Numerical results indicate that this training strategy effectively reduces computational cost without sacrificing accuracy, thereby enabling the efficient solution of complex multi-scale PDE problems in both forward and inverse settings.
Target speech extraction (TSE) has achieved strong performance under relatively simple acoustic conditions, such as one-speaker-plus-noise and two-speaker mixtures. However, its performance degrades noticeably in more challenging scenarios where interfering speakers and background noise coexist, particularly for embedding-free TSE methods, due to noise-corrupted target guidance arising from enrollment-mixture interaction as well as increased speech distortion. To address this issue, we introduce a lightweight speech enhancement model, GTCRN, to provide more reliable guidance for TSE under such challenging acoustic conditions. Building upon our previously proposed competitive embedding-free framework SEF-PNet, we propose two enhanced systems, termed LGTSE and D-LGTSE. LGTSE incorporates noise-suppressed guidance by denoising the noisy mixture before performing context interaction with the clean enrollment speech, thereby reducing the adverse impact of noise during guidance construction. D-LGTSE further improves robustness against speech distortion by exploiting the denoised outputs as additional distorted training data, which enriches acoustic variability and enables the model to better handle distorted inputs. In addition, a multi-stage training strategy is adopted, where the enhancement module is first pre-trained, followed by training the TSE backbone, and finally joint fine-tuning of the entire system to fully exploit their complementarity. Experiments on the Libri2Mix and WHAM! datasets demonstrate consistent improvements with only negligible additional computational cost. On Libri2Mix, D-LGTSE achieves gains of 0.89 dB in SI-SDR, 0.16 in PESQ, and 1.97% in STOI over the baseline; on WHAM!, it delivers improvements of 0.50 dB, 0.03, and 0.59%. When further equipped with the stronger CIE-mDPTNet backbone, D-LGTSE-mDPTNet delivers even larger gains of 1.30 dB, 0.12, and 1.68% on WHAM!, confirming the generalizability across different backbones and noise conditions. Our code is publicly available at https://github.com/isHuangZiling/D-LGTSE.
Model correction is essential for reliable PDE learning under misspecified physics arising from simplified assumptions or limited observations. Most existing correction methods operate in parameter space, where uncertainty quantification typically requires sampling or ensembles and can be prohibitively expensive. To address this challenge, we propose a representation-level, latent-space correction framework by extending LVM-GP, a probabilistic PDE solver that couples a latent variable model with Gaussian processes. Our method uses a shared confidence-aware encoder with two probabilistic decoders to jointly model the solution and a discrepancy correction. The encoder learns a stochastic latent representation by interpolating between deterministic features and a GP prior via a learnable confidence function. Conditioned on this shared latent representation, the two decoders jointly quantify uncertainty in both the solution and the correction under soft physics constraints with noisy data. A latent-space regularizer further structures the representation and improves robustness. This unified design enables joint uncertainty quantification of the solution and the correction within a single training procedure, without parameter sampling or repeated retraining. Numerical experiments show accuracy comparable to Ensemble PINNs and B-PINNs, with improved computational efficiency and robustness to misspecified physics.
A complete classification is established for all continuous and SL(n) covariant higher-order symmetric-tensor-valued valuations on Lq-spaces. They are shown to be essentially the class of moment tensors.
Let F be a graph with chromatic number χ(F) = r+1. Denote by ex(n, F) and Ex(n, F) the Turán number and the set of all extremal graphs for F, respectively. In addition, ex_ssp(n, F) and Ex_ssp(n, F) are the maximum signless Laplacian spectral radius of all n-vertex F-free graphs and the set of all n-vertex F-free graphs with signless Laplacian spectral radius ex_ssp(n, F), respectively. It is known that Ex_ssp(n, F)⊃ Ex(n, F) if F is a triangle. In this paper, employing the regularity method and Füredi's stability theorem, we prove that for a given graph F and r⩾ 3, if ex(n, F) = t_r(n)+O(1), then Ex_ssp(n, F) ⊆ Ex(n, F) for sufficiently large n, where t_r(n) is the number of edges in the Turán graph T_r(n).
