fTNN：用于分数阶偏微分方程的张量神经网络
fTNN: a tensor neural network for fractional PDEs
AI总结：
提出fTNN，一种确定性张量神经网络子空间方法，通过几何自适应积分分裂和边界奇异性感知试函数求解分数阶拉普拉斯问题，在强边界奇异性和长时间模拟中显著优于fPINN和蒙特卡洛方法。
摘要：
我们开发了fTNN，一种用于有界域上涉及分数阶拉普拉斯问题的确定性张量神经网络子空间方法，以分数阶泊松方程和含时分数阶对流扩散方程为典型代表。该方法采用几何自适应积分分裂，包含空间依赖的近场半径，将分数阶拉普拉斯算子分解为三个贡献：奇异近场、规则内部远场和解析外部远场。然后，奇异径向积分采用Gauss-Jacobi求积，规则径向积分采用Gauss求积，角度变量采用确定性角度求积，从而得到分数阶拉普拉斯算子的完全确定性积分框架。为了精确求解低正则性解及其相关损失泛函，我们构造了边界奇异性感知的试函数，该函数显式包含边界特征，并提出了两种策略来自动选择主导指数并根据分数阶算子或分数阶算子与源项共同引起的奇异性结构评估损失函数。对于含时分数阶偏微分方程，我们设计了一种时空可分离的神经网络，将时空残差分解为低维时间积分和空间积分之和，并将该表示与交替神经网络子空间优化策略相结合以实现高效训练。数值实验表明，所提出的框架在测试基准上达到了高精度，并且显著优于现有的fPINN和蒙特卡洛基线，特别是在强边界奇异性和长时间模拟问题中。
英文摘要：
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.
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