FPDE

class pinnx.problem.FPDE(geometry, pde, alpha, constraints, resolution, approximator=None, meshtype='dynamic', num_domain=0, num_boundary=0, train_distribution='Hammersley', anchors=None, solution=None, num_test=None, loss_fn='MSE', loss_weights=None)

Fractional PDE solver.

D-dimensional fractional Laplacian of order alpha/2 (1 < alpha < 2) is defined as: (-Delta)^(alpha/2) u(x) = C(alpha, D) int_{||theta||=1} D_theta^alpha u(x) d theta, where C(alpha, D) = gamma((1-alpha)/2) * gamma((D+alpha)/2) / (2 pi^((D+1)/2)), D_theta^alpha is the Riemann-Liouville directional fractional derivative, and theta is the differentiation direction vector. The solution u(x) is assumed to be identically zero in the boundary and exterior of the domain. When D = 1, C(alpha, D) = 1 / (2 cos(alpha * pi / 2)).

This solver does not consider C(alpha, D) in the fractional Laplacian, and only discretizes int_{||theta||=1} D_theta^alpha u(x) d theta. D_theta^alpha is approximated by Grunwald-Letnikov formula.

References

G. Pang, L. Lu, & G. E. Karniadakis. fPINNs: Fractional physics-informed neural networks. SIAM Journal on Scientific Computing, 41(4), A2603–A2626, 2019.

test()

Return a test dataset.

train_next_batch(batch_size=None)

Return a training dataset of the size batch_size.