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

Hillebrecht, Birgit; Unger, Benjamin

doi:10.48550/arxiv.2203.17055

Cited by 4 publications

(4 citation statements)

References 16 publications

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“…However, it is highly challenging since the neural networks used in this paper include multiple different layers. We refer the interested readers to [14,20,26,27,30] on this topic.…”

Section: Discussionmentioning

confidence: 99%

Divide-and-conquer DNN approach for the inverse point source problem using a few single frequency measurements

Du,

Li,

Liu

et al. 2023

Inverse Problems

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We consider the inverse problem to determine the number and locations of acoustic point sources from single low-frequency partial data. The problem is particularly challenging in the sense that the data is available only at a few locations which span a small aperture. Integrating the deep neural networks (DNNs) and Bayesian inversion, we propose a divide-and-conquer approach by dividing the inverse problem into three subproblems. The first subproblem is to determine the number of point sources, which is formulated as a common machine learning task—classification. A simple DNN is proposed and trained to predict the numbers of the point sources. The second subproblem is to reconstruct the (approximate) locations of the point sources. We formulate the problem as a nonlinear function with the input being the measured data and the output being a carefully elaborated location vector. Then a second DNN is proposed to learn the mapping and predict the location vector effectively. The location vector is post-processed to provide an indicator (image) function for the (approximate) locations of the point sources. The third subproblem is to improve the accuracy of the location prediction, for which we employ a Bayesian inversion algorithm. This divide-and-conquer approach can effectively treat both phase and phaseless data as demonstrated by various examples.

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“…However, it is highly challenging since the neural networks used in this paper include multiple different layers. We refer the interested readers to [14,20,26,27,30] on this topic.…”

Section: Discussionmentioning

confidence: 99%

Divide-and-conquer DNN approach for the inverse point source problem using a few single frequency measurements

Du,

Li,

Liu

et al. 2023

Inverse Problems

View full text Add to dashboard Cite

show abstract

“…Other theoretical analyses of PINNs include e.g. [67,68,27]. For DeepONets, convergence rates for advection-diffusion equations are presented in [15] and a clear workflow for obtaining generic error estimates as well as worked out examples can be found in [40].…”

Section: Related Work and Discussionmentioning

confidence: 99%

Generic bounds on the approximation error for physics-informed (and) operator learning

Ryck¹,

Mishra²

2022

Preprint

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We propose a very general framework for deriving rigorous bounds on the approximation error for physics-informed neural networks (PINNs) and operator learning architectures such as DeepONets and FNOs as well as for physics-informed operator learning. These bounds guarantee that PINNs and (physics-informed) Deep-ONets or FNOs will efficiently approximate the underlying solution or solution operator of generic partial differential equations (PDEs). Our framework utilizes existing neural network approximation results to obtain bounds on more involved learning architectures for PDEs. We illustrate the general framework by deriving the first rigorous bounds on the approximation error of physics-informed operator learning and by showing that PINNs (and physics-informed DeepONets and FNOs) mitigate the curse of dimensionality in approximating nonlinear parabolic PDEs.Preprint. Under review.

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“…Although the previous studies have described some of the difficulties and dynamics of the training phase, there is no universal theory on convergence rate or a priori measure of success of the training, and the present error estimators need to be tightened for practical non-linear systems [20]. In this paper, we provide some evidence that connects the dimensionality of solution, in the Kolmogorov n-width sense, to the difficulties in the training phase.…”

Section: Introductionmentioning

confidence: 93%

Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks

Mojgani,

Balajewicz,

Hassanzadeh

2022

Preprint

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Physics-informed neural networks (PINNs) leverage neural-networks to find the solutions of partial differential equation (PDE)-constrained optimization problems with initial conditions and boundary conditions as soft constraints. These soft constraints are often considered to be the sources of the complexity in the training phase of PINNs. Here, we demonstrate that the challenge of training (i) persists even when the boundary conditions are strictly enforced, and (ii) is closely related to the Kolmogorov n-width associated with problems demonstrating transport, convection, traveling waves, or moving fronts. Given this realization, we describe the mechanism underlying the training schemes such as those used in eXtended PINNs (XPINN), curriculum regularization, and sequence-to-sequence learning. For an important category of PDEs, i.e., governed by non-linear convection-diffusion equation, we propose reformulating PINNs on a Lagrangian frame of reference, i.e., LPINNs, as a PDE-informed solution. A parallel architecture with two branches is proposed. One branch solves for the state variables on the characteristics, and the second branch solves for the low-dimensional characteristics curves. The proposed architecture conforms to the causality innate to the convection, and leverages the direction of travel of the information in the domain. Finally, we demonstrate that the loss landscapes of LPINNs are less sensitive to the so-called "complexity" of the problems, compared to those in the traditional PINNs in the Eulerian framework.

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

Cited by 4 publications

References 16 publications

Divide-and-conquer DNN approach for the inverse point source problem using a few single frequency measurements

Divide-and-conquer DNN approach for the inverse point source problem using a few single frequency measurements

Generic bounds on the approximation error for physics-informed (and) operator learning

Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks

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