48,976 research outputs found
Operator Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations Characterized by Sharp Solutions
Physics-informed Neural Networks (PINNs) have been shown as a promising
approach for solving both forward and inverse problems of partial differential
equations (PDEs). Meanwhile, the neural operator approach, including methods
such as Deep Operator Network (DeepONet) and Fourier neural operator (FNO), has
been introduced and extensively employed in approximating solution of PDEs.
Nevertheless, to solve problems consisting of sharp solutions poses a
significant challenge when employing these two approaches. To address this
issue, we propose in this work a novel framework termed Operator Learning
Enhanced Physics-informed Neural Networks (OL-PINN). Initially, we utilize
DeepONet to learn the solution operator for a set of smooth problems relevant
to the PDEs characterized by sharp solutions. Subsequently, we integrate the
pre-trained DeepONet with PINN to resolve the target sharp solution problem. We
showcase the efficacy of OL-PINN by successfully addressing various problems,
such as the nonlinear diffusion-reaction equation, the Burgers equation and the
incompressible Navier-Stokes equation at high Reynolds number. Compared with
the vanilla PINN, the proposed method requires only a small number of residual
points to achieve a strong generalization capability. Moreover, it
substantially enhances accuracy, while also ensuring a robust training process.
Furthermore, OL-PINN inherits the advantage of PINN for solving inverse
problems. To this end, we apply the OL-PINN approach for solving problems with
only partial boundary conditions, which usually cannot be solved by the
classical numerical methods, showing its capacity in solving ill-posed problems
and consequently more complex inverse problems.Comment: Preprint submitted to Elsevie
On analysis error covariances in variational data assimilation
The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The equation for the analysis error is derived through the errors of the input data (background and observation errors). This equation is used to show that in a nonlinear case the analysis error covariance operator can be approximated by the inverse Hessian of an auxiliary data assimilation problem which involves the tangent linear model constraints. The inverse Hessian is constructed by the quasi-Newton BFGS algorithm when solving the auxiliary data assimilation problem. A fully nonlinear ensemble procedure is developed to verify the accuracy of the proposed algorithm. Numerical examples are presented
On optimal solution error covariances in variational data assimilation problems
The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find unknown parameters such as distributed model coefficients or boundary conditions. The equation for the optimal solution error is derived through the errors of the input data (background and observation errors), and the optimal solution error covariance operator through the input data error covariance operators, respectively. The quasi-Newton BFGS algorithm is adapted to construct the covariance matrix of the optimal solution error using the inverse Hessian of an auxiliary data assimilation problem based on the tangent linear model constraints. Preconditioning is applied to reduce the number of iterations required by the BFGS algorithm to build a quasi-Newton approximation of the inverse Hessian. Numerical examples are presented for the one-dimensional convection-diffusion model
Multigrid waveform relaxation for the time-fractional heat equation
In this work, we propose an efficient and robust multigrid method for solving
the time-fractional heat equation. Due to the nonlocal property of fractional
differential operators, numerical methods usually generate systems of equations
for which the coefficient matrix is dense. Therefore, the design of efficient
solvers for the numerical simulation of these problems is a difficult task. We
develop a parallel-in-time multigrid algorithm based on the waveform relaxation
approach, whose application to time-fractional problems seems very natural due
to the fact that the fractional derivative at each spatial point depends on the
values of the function at this point at all earlier times. Exploiting the
Toeplitz-like structure of the coefficient matrix, the proposed multigrid
waveform relaxation method has a computational cost of
operations, where is the number of time steps and is the number of
spatial grid points. A semi-algebraic mode analysis is also developed to
theoretically confirm the good results obtained. Several numerical experiments,
including examples with non-smooth solutions and a nonlinear problem with
applications in porous media, are presented
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