74 research outputs found
Optimally convergent hybridizable discontinuous Galerkin method for fifth-order Korteweg-de Vries type equations
We develop and analyze the first hybridizable discontinuous Galerkin (HDG)
method for solving fifth-order Korteweg-de Vries (KdV) type equations. We show
that the semi-discrete scheme is stable with proper choices of the
stabilization functions in the numerical traces. For the linearized fifth-order
equations, we prove that the approximations to the exact solution and its four
spatial derivatives as well as its time derivative all have optimal convergence
rates. The numerical experiments, demonstrating optimal convergence rates for
both the linear and nonlinear equations, validate our theoretical findings
A robust error estimator and a residual-free error indicator for reduced basis methods
The Reduced Basis Method (RBM) is a rigorous model reduction approach for
solving parametrized partial differential equations. It identifies a
low-dimensional subspace for approximation of the parametric solution manifold
that is embedded in high-dimensional space. A reduced order model is
subsequently constructed in this subspace. RBM relies on residual-based error
indicators or {\em a posteriori} error bounds to guide construction of the
reduced solution subspace, to serve as a stopping criteria, and to certify the
resulting surrogate solutions. Unfortunately, it is well-known that the
standard algorithm for residual norm computation suffers from premature
stagnation at the level of the square root of machine precision.
In this paper, we develop two alternatives to the standard offline phase of
reduced basis algorithms. First, we design a robust strategy for computation of
residual error indicators that allows RBM algorithms to enrich the solution
subspace with accuracy beyond root machine precision. Secondly, we propose a
new error indicator based on the Lebesgue function in interpolation theory.
This error indicator does not require computation of residual norms, and
instead only requires the ability to compute the RBM solution. This
residual-free indicator is rigorous in that it bounds the error committed by
the RBM approximation, but up to an uncomputable multiplicative constant.
Because of this, the residual-free indicator is effective in choosing snapshots
during the offline RBM phase, but cannot currently be used to certify error
that the approximation commits. However, it circumvents the need for \textit{a
posteriori} analysis of numerical methods, and therefore can be effective on
problems where such a rigorous estimate is hard to derive
GPT-PINN: Generative Pre-Trained Physics-Informed Neural Networks toward non-intrusive Meta-learning of parametric PDEs
Physics-Informed Neural Network (PINN) has proven itself a powerful tool to
obtain the numerical solutions of nonlinear partial differential equations
(PDEs) leveraging the expressivity of deep neural networks and the computing
power of modern heterogeneous hardware. However, its training is still
time-consuming, especially in the multi-query and real-time simulation
settings, and its parameterization often overly excessive. In this paper, we
propose the Generative Pre-Trained PINN (GPT-PINN) to mitigate both challenges
in the setting of parametric PDEs. GPT-PINN represents a brand-new
meta-learning paradigm for parametric systems. As a network of networks, its
outer-/meta-network is hyper-reduced with only one hidden layer having
significantly reduced number of neurons. Moreover, its activation function at
each hidden neuron is a (full) PINN pre-trained at a judiciously selected
system configuration. The meta-network adaptively ``learns'' the parametric
dependence of the system and ``grows'' this hidden layer one neuron at a time.
In the end, by encompassing a very small number of networks trained at this set
of adaptively-selected parameter values, the meta-network is capable of
generating surrogate solutions for the parametric system across the entire
parameter domain accurately and efficiently
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