Solving PDEs by Variational Physics-Informed Neural Networks: an a posteriori error analysis

Berrone, Stefano; Canuto, Claudio; Pintore, Moreno

doi:10.48550/arxiv.2205.00786

Cited by 2 publications

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“…On the contrary, rigorous a posteriori error analyses are already available (see, for instance, [15]). Recently (see [16]) we derived a posteriori error estimates for the discretization setting considered in this paper.…”

Section: Introductionmentioning

confidence: 99%

Variational Physics Informed Neural Networks: the Role of Quadratures and Test Functions

2022

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In this work we analyze how quadrature rules of different precisions and piecewise polynomial test functions of different degrees affect the convergence rate of Variational Physics Informed Neural Networks (VPINN) with respect to mesh refinement, while solving elliptic boundary-value problems. Using a Petrov-Galerkin framework relying on an inf-sup condition, we derive an a priori error estimate in the energy norm between the exact solution and a suitable high-order piecewise interpolant of a computed neural network. Numerical experiments confirm the theoretical predictions and highlight the importance of the inf-sup condition. Our results suggest, somehow counterintuitively, that for smooth solutions the best strategy to achieve a high decay rate of the error consists in choosing test functions of the lowest polynomial degree, while using quadrature formulas of suitably high precision.

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Section: Introductionmentioning

confidence: 99%

Variational Physics Informed Neural Networks: the Role of Quadratures and Test Functions

2022

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“…In distinction, the error estimator presented here is computable which we demonstrate with several academic examples. Other earlier works [5], [31] also differ from the following work in fundamental aspects, such as that the error bounds derived in them are either not guaranteed to hold, are restricted to a certain problem type, or require discretization steps.…”

Section: Introductionmentioning

confidence: 99%

Certified machine learning: A posteriori error estimation for physics-informed neural networks

Hillebrecht

Unger

2022

2022 International Joint Conference on Neural Networks (IJCNN)

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Prediction error quantification in machine learning has been left out of most methodological investigations of neural networks, for both purely data-driven and physics-informed approaches. Beyond statistical investigations and generic results on the approximation capabilities of neural networks, we present a rigorous upper bound on the prediction error of physicsinformed neural networks. This bound can be calculated without the knowledge of the true solution and only with a priori available information about the characteristics of the underlying dynamical system governed by a partial differential equation. We apply this a posteriori error bound exemplarily to four problems: the transport equation, the heat equation, the Navier-Stokes equation and the Klein-Gordon equation.

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Solving PDEs by Variational Physics-Informed Neural Networks: an a posteriori error analysis

Cited by 2 publications

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Variational Physics Informed Neural Networks: the Role of Quadratures and Test Functions

Variational Physics Informed Neural Networks: the Role of Quadratures and Test Functions

Certified machine learning: A posteriori error estimation for physics-informed neural networks

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