相关论文: SpectraKAN: Conditioning Spectral Operators

Microstructural evolution, particularly grain growth, plays a critical role in shaping the physical, optical, and electronic properties of materials. Traditional phase-field modeling accurately simulates these phenomena but is…

Partial differential equations (PDEs) are fundamental for modeling complex physical systems, yet classical numerical solvers face prohibitive computational costs in high-dimensional and multi-scale regimes. While Transformer-based neural…

We introduce sparse autoencoder neural operators (SAE-NOs), a new class of sparse autoencoders that operate in function spaces rather than fixed-dimensional Euclidean representations. We formalize the functional representation hypothesis,…

We investigate the spectrum of differentiation matrices for certain operators on the sphere that are generated from collocation at a set of scattered points $X$ with positive definite and conditionally positive definite kernels. We focus on…

This paper introduces significant advancements in fractional neural operators (FNOs) through the integration of adaptive hybrid kernels and stochastic multiscale analysis. We address several open problems in the existing literature by…

Spectral Barron spaces, constituting a specialized class of function spaces that serve as an interdisciplinary bridge between mathematical analysis, partial differential equations (PDEs), and machine learning, are distinguished by the decay…

Spectral methods employing non-standard polynomial bases, such as M\"untz polynomials, have proven effective for accurately solving problems with solutions exhibiting low regularity, notably including sub-diffusion equations. However, due…

This paper proposes a novel neural network framework, denoted as spectral integrated neural networks (SINNs), for resolving three-dimensional forward and inverse dynamic problems. In the SINNs, the spectral integration method is applied to…

Fourier Neural Operators (FNOs) have been promoted as fast, mesh-invariant surrogates for partial-differential equation solvers, with seismic studies reporting orders-of-magnitude speedup over classical methods. We revisit those claims by…

In this paper, we propose a refinement strategy to the well-known Physics-Informed Neural Networks (PINNs) for solving partial differential equations (PDEs) based on the concept of Optimal Transport (OT). Conventional black-box PINNs…

Simulation tools for photoacoustic wave propagation have played a key role in advancing photoacoustic imaging by providing quantitative and qualitative insights into parameters affecting image quality. Classical methods for numerically…

Frequency synthesis and spectro-temporal control of optical wave packets are central to ultrafast science, with supercontinuum (SC) generation standing as one remarkable example. Through passive manipulation, femtosecond (fs) pulses from…

Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural…

Neural operators have emerged as powerful tools for learning solution operators of partial differential equations. However, in time-dependent problems, standard training strategies such as teacher forcing introduce a mismatch between…

The lacking of analytic solutions of diverse partial differential equations (PDEs) gives birth to a series of computational techniques for numerical solutions. Although numerous latest advances are accomplished in developing neural…

Solving parametric Partial Differential Equations (PDEs) for a broad range of parameters is a critical challenge in scientific computing. To this end, neural operators, which \textcolor{black}{predicts the PDE solution with variable PDE…

Spectral behaviors have been widely discussed in machine learning, yet the optimizer's own spectral bias remains unclear. We argue that first-order optimizers exhibit an intrinsic frequency preference that significantly reshapes the…

Physics-Informed Neural Networks with hard constraints (HC-PINNs) are increasingly favored for their ability to strictly enforce boundary conditions via a trial function ansatz $\tilde{u} = A + B \cdot N$, yet the theoretical mechanisms…

Kolmogorov-Arnold networks (KANs) represent data features by learning the activation functions and demonstrate superior accuracy with fewer parameters, FLOPs, GPU memory usage (Memory), shorter training time (TraT), and testing time (TesT)…

Learning complex dynamics driven by partial differential equations directly from data holds great promise for fast and accurate simulations of complex physical systems. In most cases, this problem can be formulated as an operator learning…