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On generalization error estimates of physics informed neural networks for approximating dispersive PDEs
(2021)SAM Research ReportPhysics informed neural networks (PINNs) have recently been widely used for robust and accurate approximation of PDEs. We provide rigorous upper bounds on the generalization error of PINNs approximating solutions of the forward problem for several dispersive PDEs.Report -
Numerical approximation of statistical solutions of incompressible flow
(2015)Research ReportWe present a finite difference-(Multi-level) Monte Carlo algorithm to efficiently compute statistical solutions of the two dimensional Navier-Stokes equations, with periodic bound- ary conditions and for arbitrarily high Reynolds number. We propose a reformulation of statistical solutions in the vorticity-stream function form. The vorticity-stream function for- mulation is discretized with a finite difference scheme. We obtain a convergence ...Report -
Efficient computation of all speed flows using an entropy stable shock-capturing space-time discontinuous Galerkin method
(2014)Research ReportWe present a shock-capturing space-time Discontinuous Galerkin method to approximate all speed flows modeled by systems of conservation laws with multiple time scales. The method provides a very general and computationally efficient framework for approximating such systems on account of its ability to incorporate large time steps. Numerical examples ranging from computing the incompressible limit (robustness with respect to Mach number) ...Report -
On universal approximation and error bounds for Fourier Neural Operators
(2021)SAM Research ReportFourier neural operators (FNOs) have recently been proposed as an effective framework for learning operators that map between infinite-dimensional spaces. We prove that FNOs are universal, in the sense that they can approximate any continuous operator to desired accuracy. Moreover, we suggest a mechanism by which FNOs can approximate operators associated with PDEs efficiently. Explicit error bounds are derived to show that the size of the ...Report -
On the well-posedness of Bayesian inversion for PDEs with ill-posed forward problems
(2021)SAM Research ReportWe study the well-posedness of Bayesian inverse problems for PDEs, for which the underlying forward problem may be ill-posed. Such PDEs, which include the fundamental equations of fluid dynamics, are characterized by the lack of rigorous global existence and stability results as well as possible non-convergence of numerical approximations. Under very general hypotheses on approximations to these PDEs, we prove that the posterior measure, ...Report -
Well-posedness of Bayesian inverse problems for hyperbolic conservation laws
(2021)SAM Research ReportWe study the well-posedness of the Bayesian inverse problem for scalar hyperbolic conservation laws where the statistical information about inputs such as the initial datum and (possibly discontinuous) flux function are inferred from noisy measurements. In particular, the Lipschitz continuity of the measurement to posterior map as well as the stability of the posterior to approximations, are established with respect to the Wasserstein ...Report -
Multi-level Monte Carlo finite volume method for shallow water equations with uncertain parameters applied to landslides-generated tsunamis
(2014)SAM Research ReportTwo layer Savage-Hutter type shallow water PDEs model flows such as tsunamis generated by rockslides. On account of heterogeneities in the composition of the granular matter, these models contain uncertain parameters like the ratio of densities of layers, Coulomb and interlayer friction. These parameters are modeled statistically and quantifying the resulting solution uncertainty (UQ) is a crucial task in geophysics. We propose a novel ...Report -
Long Expressive Memory for Sequence Modeling
(2021)SAM Research ReportWe propose a novel method called Long Expressive Memory (LEM) for learning long-term sequential dependencies. LEM is gradient-based, it can efficiently process sequential tasks with very long-term dependencies, and it is sufficiently expressive to be able to learn complicated input-output maps. To derive LEM, we consider a system of multiscale ordinary differential equations, as well as a suitable time-discretization of this system. For ...Report -
ENO reconstruction and ENO interpolation are stable
(2011)SAM Research ReportWe prove stability estimates for the ENO reconstruction and ENO interpolation procedures. In particular, we show that the jump of the reconstructed ENO pointvalues at each cell interface has the same sign as the jump of the underlying cell averages across that interface. We also prove that the jump of the reconstructed values can be upper-bounded in terms of the jump of the underlying cell averages. Similar sign properties hold for the ...Report -
Entropy stable schemes for initial-boundary-value conservation laws
(2010)SAM Research ReportWe consider initial boundary value problems for systems of conservation laws and design entropy stable finite difference schemes to approximate them. The schemes are shown to be entropy stable for a large class of systems that are equipped with a symmetric splitting, derived from the entropy formulation. Numerical examples for the Euler equations of gas dynamics are presented to illustrate the robust performance of the proposed method.Report