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Neural operators for time-dependent PDEs face a structural tension: spectral architectures (FNO and descendants) inherit exponential rollout-error growth from their one-step Lipschitz constant, while hierarchical U-Net operators trade…

Neural operators have achieved strong performance in learning solution operators of partial differential equations (PDEs), but their inherently continuous representations struggle to capture discontinuities and sharp transitions. Existing…

Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training…

In this paper, we introduce the Spectral Coefficient Learning via Operator Network (SCLON), a novel operator learning-based approach for solving parametric partial differential equations (PDEs) without the need for data harnessing. The…

Current Point-based detectors can only learn from the provided points, with limited receptive fields and insufficient global learning capabilities for such targets. In this paper, we present a novel Point Dilation Mechanism for single-stage…

We develop the spectral edge analysis: phase transitions in neural network training -- grokking, capability gains, loss plateaus -- are controlled by the spectral gap of the rolling-window Gram matrix of parameter updates. In the extreme…

In recent years, Neural Operators(NO) have gradually emerged as a popular approach for solving Partial Differential Equations (PDEs). However, their application to large-scale engineering tasks suffers from significant computational…

Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the…

Unlike ODEs, whose models involve system matrices and whose controllers involve vector or matrix gains, PDE models involve functions in those roles functional coefficients, dependent on the spatial variables, and gain functions dependent on…

Solving partial differential equations (PDEs) by numerical methods meet computational cost challenge for getting the accurate solution since fine grids and small time steps are required. Machine learning can accelerate this process, but…

Most neural-operator surrogates for PDEs inherit from DeepONet-style formulations the requirement that the input function be sampled at a fixed, ordered set of sensors. This assumption limits applicability to problems with variable sensor…

Accurate modeling of spatiotemporal dynamics is crucial to understanding complex phenomena across science and engineering. However, this task faces a fundamental challenge when the governing equations are unknown and observational data are…

Modeling high-frequency information is a critical challenge in scientific machine learning. For instance, fully turbulent flow simulations of the Navier-Stokes equations at Reynolds numbers 3500 and above can generate high-frequency signals…

Autoregressive neural PDE surrogates predict future states by repeatedly applying a learned one-step operator. This is a simple and widely used method, but small one-step errors can accumulate during long rollouts. The resulting drift often…

The brain's spatial orientation system uses different neuron ensembles to aid in environment-based navigation. Two of the ways brains encode spatial information is through head direction cells and grid cells. Brains use head direction cells…

Deep Operator Networks (DeepONets) have become a central tool in data-driven operator learning, providing flexible surrogates for nonlinear mappings arising in partial differential equations (PDEs). However, the standard trunk design based…

In this paper, we focus on fraud detection on a signed graph with only a small set of labeled training data. We propose a novel framework that combines deep neural networks and spectral graph analysis. In particular, we use the node…

Many real-world relational systems, such as social networks and biological systems, contain dynamic interactions. When learning dynamic graph representation, it is essential to employ sequential temporal information and geometric structure.…

Spectral bias implies an imbalance in training dynamics, whereby high-frequency components may converge substantially more slowly than low-frequency ones. To alleviate this issue, we propose a cross-attention-based architecture that…

The memory consumption of most Convolutional Neural Network (CNN) architectures grows rapidly with increasing depth of the network, which is a major constraint for efficient network training on modern GPUs with limited memory, embedded…