2 research outputs found
A spectral penalty method for two-sided fractional differential equations with general boundary conditions
We consider spectral approximations to the conservative form of the two-sided
Riemann-Liouville (R-L) and Caputo fractional differential equations (FDEs)
with nonhomogeneous Dirichlet (fractional and classical, respectively) and
Neumann (fractional) boundary conditions. In particular, we develop a spectral
penalty method (SPM) by using the Jacobi poly-fractonomial approximation for
the conservative R-L FDEs while using the polynomial approximation for the
conservative Caputo FDEs. We establish the well-posedness of the corresponding
weak problems and analyze sufficient conditions for the coercivity of the SPM
for different types of fractional boundary value problems. This analysis allows
us to estimate the proper values of the penalty parameters at boundary points.
We present several numerical examples to verify the theory and demonstrate the
high accuracy of SPM, both for stationary and time dependent FDEs. Moreover, we
compare the results against a Petrov-Galerkin spectral tau method (PGS-,
an extension of [Z. Mao, G.E. Karniadakis, SIAM J. Numer. Anal., 2018]) and
demonstrate the superior accuracy of SPM for all cases considered.Comment: 27 page
Variational Physics-Informed Neural Networks For Solving Partial Differential Equations
Physics-informed neural networks (PINNs) [31] use automatic differentiation
to solve partial differential equations (PDEs) by penalizing the PDE in the
loss function at a random set of points in the domain of interest. Here, we
develop a Petrov-Galerkin version of PINNs based on the nonlinear approximation
of deep neural networks (DNNs) by selecting the {\em trial space} to be the
space of neural networks and the {\em test space} to be the space of Legendre
polynomials. We formulate the \textit{variational residual} of the PDE using
the DNN approximation by incorporating the variational form of the problem into
the loss function of the network and construct a \textit{variational
physics-informed neural network} (VPINN). By integrating by parts the integrand
in the variational form, we lower the order of the differential operators
represented by the neural networks, hence effectively reducing the training
cost in VPINNs while increasing their accuracy compared to PINNs that
essentially employ delta test functions. For shallow networks with one hidden
layer, we analytically obtain explicit forms of the \textit{variational
residual}. We demonstrate the performance of the new formulation for several
examples that show clear advantages of VPINNs over PINNs in terms of both
accuracy and speed.Comment: 24 pages, 12 figure