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- Euler-Gleichungen, isotherme Euler-Gleichungen, Modellhierarchie, Netzelemente (1)
- Flow Networks (1)
- Gas Networks (1)
- Kernel Density Estimator (1)
- Probabilistic Constraints (1)
- Spheric Radial Decomposition (1)
- error estimators (1)
- gas supply networks (1)
- model adaptivity (1)
- model hierarchy (1)
- refinement strategies (1)
Physics informed neural networks have been recently proposed and offer a new promising method to solve differential equations. They have been adapted to many more scenarios and different variations of the original method have been proposed. In this case study we review many of these variations. We focus on variants that can compensate for imbalances in the loss function and perform a comprehensive numerical comparison of these variants with application to gas transport problems. Our case study includes different formulations of the loss function, different algorithmic loss balancing methods, different optimization schemes and different numbers of parameters and sampling points. We conclude that the original PINN approach with specifically chosen constant weights in the loss function gives the best results in our tests. These weights have been obtained by a computationally expensive random-search scheme. We further conclude for our test case that loss balancing methods which were developed for other differential equations have no benefit for gas transport problems, that the control volume physics informed formulation has no benefit against the initial formulation and that the best optimization strategy is the L-BFGS method.
In this paper, we are concerned with the quantification of
uncertainties that arise from intra-day oscillations in the
demand for natural gas transported through large-scale networks. The
short-term transient dynamics of the gas flow is modelled by a hierarchy
of hyperbolic systems of balance laws based on the isentropic Euler equations.
We extend a novel adaptive strategy for solving elliptic PDEs with random data,
recently proposed and analysed by Lang, Scheichl, and Silvester
[J. Comput. Phys., 419:109692, 2020], to uncertain gas transport problems.
Sample-dependent adaptive meshes and a model refinement in the physical space
is combined with adaptive anisotropic sparse Smolyak grids in the stochastic
space. A single-level approach which balances the discretization errors of
the physical and stochastic approximations and
a multilevel approach which additionally minimizes the computational costs
are considered. Two examples taken from a public gas library demonstrate the
reliability of the error control of expectations calculated from
random quantities of interest, and the further use of
stochastic interpolants to, e.g., approximate probability density functions
of minimum and maximum pressure values at the exits of the network.
High-dimensional interpolation problems appear in various applications of uncertainty quantification, stochastic optimization and machine learning. Such problems are computationally expensive and request the use of adaptive grid generation strategies like anisotropic sparse grids to mitigate the curse of dimensionality. However, it is well known that the standard dimension-adaptive sparse grid method converges very slowly or even fails in the case of non-smooth functions. For piecewise smooth functions with kinks, we construct two novel hp-adaptive sparse grid collocation algorithms that combine low-order basis functions with local support in parts of the domain with less regularity and variable-order basis functions elsewhere. Spatial refinement is realized by means of a hierarchical multivariate knot tree which allows the construction of localised hierarchical basis functions with varying order. Hierarchical surplus is used as an error indicator to automatically detect the non-smooth region and adaptively refine the collocation points there. The local polynomial degrees are optionally selected by a greedy approach or a kink detection procedure. Three numerical benchmark examples with different dimensions are discussed and comparison with locally linear and highest degree basis functions are given to show the efficiency and accuracy of the proposed methods.
Method-of-lines discretizations are demanding test problems for stiff inte-
gration methods. However, for PDE problems with known analytic solution
the presence of space discretization errors or the need to use codes to compute
reference solutions may limit the validity of numerical test results. To over-
come these drawbacks we present in this short note a simple test problem with
boundary control, a situation where one-step methods may suffer from order
reduction. We derive exact formulas for the solution of an optimal boundary
control problem governed by a one-dimensional discrete heat equation and an
objective function that measures the distance of the final state from the target
and the control costs. This analytical setting is used to compare the numeri-
cally observed convergence orders for selected implicit Runge-Kutta and Peer
two-step methods of classical order four which are suitable for optimal control
problems.