fTNN: a tensor neural network for fractional PDEs
We develop the fTNN, a deterministic tensor neural network subspace method for problems involving the fractional Laplacian on bounded domains, taking the fractional Poisson equation and time-dependent fractional advection-diffusion equation as typical representatives. The work employs a geometry-adapted integration split featuring a spatially dependent near-field radius, which decomposes the fractional Laplacian into three contributions: a singular near field, a regular interior far field, and an analytical exterior far field. Then the singular radial integrals are treated by Gauss-Jacobi quadrature, the regular radial integrals by Gauss quadrature, and the angular variables by deterministic angular quadrature, yielding a fully deterministic integration framework of the fractional Laplacian operator. To accurately resolve low-regularity solutions and the associated loss functional, we construct boundary-singularity-aware trial functions enriched with explicit boundary features, and propose two strategies for automatically selecting the leading exponent and evaluating the loss function from the singularity structure induced by the fractional operator, or jointly by the fractional operator and the source term. For time-dependent fractional PDEs, we design a spatiotemporally separable neural network that factorizes the time-space residual into a sum of low-dimensional temporal and spatial integrals, and we integrate this representation with an alternating neural network subspace optimization strategy for efficient training. Numerical experiments show that the proposed framework attains high accuracy on the tested benchmarks and improves substantially over existing fPINN and Monte Carlo baselines, particularly for problems with strong boundary singularities and long-time simulations.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Tackling the Curse of Dimensionality in Fractional and Tempered Fractional PDEs with Physics-Informed Neural Networks
Fractional and tempered fractional partial differential equations (PDEs) are effective models of long-range interactions, anomalous diffusion, and non-local effects. Traditional numerical methods for these problems are m…
Monte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equations
We introduce a sampling based machine learning approach, Monte Carlo physics informed neural networks (MC-PINNs), for solving forward and inverse fractional partial differential equations (FPDEs). As a generalization of …
GMC-PINNs: A new general Monte Carlo PINNs method for solving fractional partial differential equations on irregular domains
Physics-Informed Neural Networks (PINNs) have been widely used for solving partial differential equations (PDEs) of different types, including fractional PDEs (fPDES) [29]. Herein, we propose a new general (quasi) Monte …
Computational EfficiencyfPINN-DeepONet: A Physics-Informed Operator Learning Framework for Multi-term Time-fractional Mixed Diffusion-wave Equations
In this paper, we develop a physics-informed deep operator learning framework for solving multi-term time-fractional mixed diffusion-wave equations (TFMDWEs). We begin by deriving an $L_2$ approximation, which achieves f…
A Multidimensional Graph Fourier Transformation Neural Network for Vehicle Trajectory Prediction
This work introduces the multidimensional Graph Fourier Transformation Neural Network (GFTNN) for long-term trajectory predictions on highways. Similar to Graph Neural Networks (GNNs), the GFTNN is a novel network archit…
DecoderDescriptiveTrajectory Prediction