Neural operators have shown great potential in solving a family of Partial Differential Equations (PDEs) by modeling the mappings between input and output functions. Fourier Neural Operator (FNO) implements global convolutions via parameterizing the integral operators in Fourier space. However, it often results in over-smoothing solutions and fails to capture local details and high-frequency components. To address these limitations, we investigate incorporating the spatial-frequency localization property of Wavelet transforms into the Transformer architecture. We propose a novel Wavelet Attention (WA) module with linear computational complexity to efficiently learn locality-aware features. Building upon WA, we further develop the Spectral Attention Operator Transformer (SAOT), a hybrid spectral Transformer framework that integrates WA’s localized focus with the global receptive field of Fourier-based Attention (FA) through a gated fusion block. Experimental results demonstrate that WA significantly mitigates the limitations of FA and outperforms existing Wavelet-based neural operators by a large margin. By integrating the locality-aware and global spectral representations, SAOT achieves state-of-the-art performance on six operator learning benchmarks and exhibits strong discretization-invariant ability.

Physics-informed neural networks (PINNs) have emerged as a new learning paradigm for solving partial differential equations (PDEs) by enforcing the constraints of physical equations, boundary conditions (BCs), and initial conditions (ICs) into the loss function. Despite their successes, vanilla PINNs still suffer from poor accuracy and slow convergence due to the intractable multi-objective optimization issue. In this paper, we propose a novel Dual-Balanced PINN (DB-PINN), which dynamically adjusts loss weights by integrating inter-balancing and intra-balancing to alleviate two imbalance issues in PINNs. Inter-balancing aims to mitigate the gradient imbalance between PDE residual loss and condition-fitting losses by determining an aggregated weight that offsets their gradient distribution discrepancies. Intra-balancing acts on condition-fitting losses to tackle the imbalance in fitting difficulty across diverse conditions. By evaluating the fitting difficulty based on the loss records, intra-balancing can allocate the aggregated weight proportionally to each condition loss according to its fitting difficulty level. We further introduce a robust weight update strategy to prevent abrupt spikes and arithmetic overflow in instantaneous weight values caused by large loss variances, enabling smooth weight updating and stable training. Extensive experiments demonstrate that DB-PINN achieves significantly superior performance than those popular gradient-based weighting methods in terms of convergence speed and prediction accuracy. Our code and supplementary material are available at https://github.com/chenhong-zhou/DualBalanced-PINNs.

Physics-informed neural networks (PINNs) have emerged as promising and powerful tools for solving partial differential equations (PDEs). To enforce PINNs that yield accurate solutions to PDEs, a set of scattered spatiotemporal points, known as collocation points, are typically sampled within the computational domain. The choices of collocation points significantly impact the performance of PINNs. However, existing sampling methods primarily rely on the PDE residual, which is insufficient for solutions with steep gradients. To enhance the accuracy of PINNs, we propose a novel multi-criteria adaptive sampling (MCAS) approach to optimally select appropriate collocation points. The MCAS approach integrates three sampling criteria: PDE’s residual, the gradient of residual, and the gradient of solutions, enabling us to capture the PDE violations and the sharpness of solutions. Experimental results demonstrate that the proposed MCAS approach is not only applicable for collocation points but also for optimizing sensor placement, consistently outperforming the residual-based sampling methods.

Class imbalance has emerged as one of the major challenges for medical image segmentation. The model cascade (MC) strategy, a popular scheme, significantly alleviates the class imbalance issue via running a set of individual deep models for coarse-to-fine segmentation. Despite its outstanding performance, however, this method leads to undesired system complexity and also ignores the correlation among the models. To handle these flaws in the MC approach, we propose in this paper a light-weight deep model, i.e., the One-pass Multi-task Network (OM-Net) to solve class imbalance better than MC does, while requiring only one-pass computation for brain tumor segmentation. First, OM-Net integrates the separate segmentation tasks into one deep model, which consists of shared parameters to learn joint features, as well as task-specific parameters to learn discriminative features. Second, to more effectively optimize OM-Net, we take advantage of the correlation among tasks to design both an online training data transfer strategy and a curriculum learning-based training strategy. Third, we further propose sharing prediction results between tasks, which enables us to design a cross-task guided attention (CGA) module. By following the guidance of the prediction results provided by the previous task, CGA can adaptively recalibrate channel-wise feature responses based on the category-specific statistics. Finally, a simple yet effective post-processing method is introduced to refine the segmentation results of the proposed attention network. Extensive experiments are conducted to demonstrate the effectiveness of the proposed techniques. Most impressively, we achieve state-of-the-art performance on the BraTS 2015 testing set and BraTS 2017 online validation set. Using these proposed approaches, we also won joint third place in the BraTS 2018 challenge among 64 participating teams.The code will be made publicly available at https://github.com/chenhong-zhou/OM-Net.