Finite Difference
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Related Articles from SNS
Arbitrary high order shaped stencils for time domain finite difference schemes in seismic wave propagation
Announce Type: replace Abstract: Finite Difference Schemes are widely used in the approximation of different hyperbolic (wave-like) differential equations, and are particularly important for seismic wave modelling and its applications. Classical methods based on Taylor Series are dominant in the literature; however, it is known that these methods can suffer from excessive numerical dispersion. In this paper, we review and extend existing high-order in space finite difference schemes for...
Arbitrary high order shaped stencils for time domain finite difference schemes in seismic wave propagation
arXiv:2606.04882v1 Announce Type: new Abstract: Finite Difference Schemes are widely used in the approximation of different hyperbolic (wave-like) differential equations, and are particularly important for seismic wave modelling and its applications. Classical methods based on Taylor Series are dominant in the literature; however, it is known that these methods can suffer from excessive numerical dispersion. In this paper, we review and extend existing high-order in space finite difference...
Physics-Informed Residuals for Adaptive Mesh Refinement in Finite-Difference PDE Solvers
Announce Type: new Abstract: Classical finite-difference solvers remain reliable tools for partial differential equations, but their efficiency depends on where mesh resolution is placed. Uniform refinement can waste degrees of freedom when solution difficulty is localised near sharp gradients, fronts, oscillations, or constraint-sensitive regions. This paper studies a hybrid strategy in which a physics-informed neural network (PINN) is used not as the final solver, but as an off-grid...
Physics-Informed Residuals for Adaptive Mesh Refinement in Finite-Difference PDE Solvers
arXiv:2606.02475v2 Announce Type: replace Abstract: Classical finite-difference solvers remain reliable tools for partial differential equations, but their efficiency depends on where mesh resolution is placed. Uniform refinement can waste degrees of freedom when solution difficulty is localised near sharp gradients, fronts, oscillations, or constraint-sensitive regions. This paper studies a hybrid strategy in which a physics-informed neural network (PINN) is used not as the final solver,...
Composite B-Spline Current Deposition and Interpolation Operators for Thin-Wire Finite-Difference Time-Domain Simulations
arXiv:2605.21450v3 Announce Type: replace Abstract: Holland-Simpson thin-wire finite-difference time-domain (FDTD) simulations of obliquely oriented closed-loop antennas exhibit persistent low-frequency parasitic currents because the current-deposition operator fails to conserve charge. This deposition operator, together with an interpolation operator that samples the tangential electric field along the wire, can be realized as regularizations of distributions: the wire current is deposited...
Overcoming the Limits of Finite Difference Method; Physics-Informed Neural Network for Noisy High-Dimensional Heat Diffusion
Announce Type: new Abstract: High-dimensional transient heat diffusion under noisy boundary conditions exposes a fundamental limitation of classical numerical methods: accuracy degrades catastrophically where physical noise is unavoidable. This paper presents a Physics-Informed Neural Network (PINN) framework as a systematic solution to this problem across one, two, and three spatial dimensions, establishing clear operational regimes that redefine solver selection in noisy thermal systems....
Magnum.np.distributed: Accelerating Finite Difference Micromagnetic Simulations with Multiple GPUs
Announce Type: new Abstract: Micromagnetic simulations are essential tools in nanomagnetism and spintronics research. Although widely adopted solvers like Mumax3 and the Python-native magnum.np use GPU acceleration to improve performance, these tools are limited to single-device computation. In this work, we present the first Python-native multi-GPU micromagnetic framework by extending magnum.np with PyTorch Distributed.
Stability beyond Bounded Differences: Sharp Generalization Bounds under Finite $L_p$ Moments
arXiv:2606.06855v1 Announce Type: cross Abstract: While algorithmic stability is a central tool for understanding generalization of learning algorithms, existing high-probability guarantees typically rely on uniform boundedness or sub-Gaussian/sub-Weibull tail assumptions, which can be overly restrictive for modern settings with heavy-tailed or unbounded losses. We develop a stability-based framework that requires only a finite $L_p$ moment condition. Our first contribution is sharp...
Evaluating Operators for Acoustic Wave Simulation Correction
arXiv:2606.08711v1 Announce Type: new Abstract: Correcting numerical dispersion artifacts from Finite Difference solvers is a well-identified challenge in computational wave physics, but existing approaches evaluate only a restricted family of CNN-based architectures and have been applied exclusively to the elastic wave equation. We instantiate the Deep Finite Difference framework on two-dimensional anisotropic acoustic wave propagation, pairing a fourth-order Finite Difference proxy with a...
Cost-Aware Learning
arXiv:2604.28020v2 Announce Type: replace Abstract: We consider the problem of Cost-Aware Learning, where sampling different components of a finite-sum objective incurs different costs. The objective is to reach a target error while minimizing the total cost. We propose Cost-Aware SGD, which uses a distribution based on gradient norms and costs to sample components.