nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applications

Pang, Guofei; D’Elia, Marta; Parks, Michael L.; Karniadakis, George

doi:10.2172/1614899

Cited by 35 publications

(43 citation statements)

References 32 publications

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“…with c(x) > 0 so that the bilinear form A(u, v) in (7) has the additional term Ω c(x)u(x)v(x)dx, then that bilinear form is coercive even for floating subdomains. In this case, the design of solution methods for (41), and in particular for (47), becomes substantially simpler.…”

Section: Equivalence Of the Single-domain And Multi-domain Problemsmentioning

confidence: 99%

“…Other critical challenges are related to the uncertain nature of model parameters; in fact, modeling parameters such as δ and those characterizing the kernel, applied forces, and/or sources can be non-measurable, sparse, and/or subject to noise. Research on such topics is very active (see, e.g., [5,4,19,20,21,16,30,41,42,43,51]) but further consideration of them is beyond the scope of this work.…”

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confidence: 99%

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A general framework for substructuring-based domain decomposition methods for models having nonlocal interactions

Capodaglio¹,

D’Elia²,

Gunzburger³

et al. 2020

Preprint

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A rigorous mathematical framework is provided for a substructuring-based domaindecomposition approach for nonlocal problems that feature interactions between points separated by a finite distance. Here, by substructuring it is meant that a traditional geometric configuration for local partial differential equation problems is used in which a computational domain is subdivided into non-overlapping subdomains. In the nonlocal setting, this approach is substructuring-based in the sense that those subdomains interact with neighboring domains over interface regions having finite volume, in contrast to the local PDE setting in which interfaces are lower dimensional manifolds separating abutting subdomains. Key results include the equivalence between the global, singledomain nonlocal problem and its multi-domain reformulation, both at the continuous and discrete levels. These results provide the rigorous foundation necessary for the development of efficient solution strategies for nonlocal domain-decomposition methods.

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Section: Equivalence Of the Single-domain And Multi-domain Problemsmentioning

confidence: 99%

mentioning

confidence: 99%

A general framework for substructuring-based domain decomposition methods for models having nonlocal interactions

Capodaglio¹,

D’Elia²,

Gunzburger³

et al. 2020

Preprint

Self Cite

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“…Nowadays, implementing PINNs and their variants [76][77][78][79][80] with parametric inputs is feasible thanks to recent hardware, software, and algorithmic advancements, including: i) notable advancements in stochastic optimization [81][82][83][84], ii) computer hardware for parallel computing, i.e., graphics processing units (GPUs), and iii) computer software, i.e., automatic differentiation-capable libraries [85], such as PyTorch [86], Tensorflow [87],…”

Section: Introductionmentioning

confidence: 99%

Physics-informed neural networks for solving parametric magnetostatic problems

Beltrán-Pulido¹,

Bilionis²,

Aliprantis³

2022

Preprint

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The optimal design of magnetic devices becomes intractable using current computational methods when the number of design parameters is high. The emerging physics-informed deep learning framework has the potential to alleviate this curse of dimensionality. The objective of this paper is to investigate the ability of physics-informed neural networks to learn the magnetic field response as a function of design parameters in the context of a two-dimensional (2-D) magnetostatic problem. Our approach is as follows. We derive the variational principle for 2-D parametric magnetostatic problems, and prove the existence and uniqueness of the solution that satisfies the equations of the governing physics, i.e., Maxwell's equations. We use a deep neural network (DNN) to represent the magnetic field as a function of space and a total of ten parameters that describe geometric features and operating point conditions. We train the DNN by minimizing the physics-informed loss function

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“…It follows from (1.1) that in fractional modeling the state of a system at a point depends on the value of the state at any other point in the space; in other words, fractional models are nonlocal. Specifically, fractional operators are special instancies of more general nonlocal operators [13,17,18,24,39] of the following form:…”

Section: Introductionmentioning

confidence: 99%

“…Based on these works, applications to image processing have also been considered [6]. On the other hand, several other works tackled the identification problem with a machine learning approach; as an example, in [39,40] the authors employ physics-informed neural networks to describe the nonlocal solution and learn model parameters such as s and δ through minimization of a loss function given by the solution mismatch and the residual of the state equation. A similar approach, referred to as operator regression and also based on minimization of the equation residual, is utilized in [50] where the authors learn nonlocal kernels, including fractional type kernels, in a least-squares regression setting.…”

Section: Introductionmentioning

confidence: 99%

An optimization-based approach to parameter learning for fractional type nonlocal models

Burkovska¹,

Glusa²,

D’Elia³

2020

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Nonlocal operators of fractional type are a popular modeling choice for applications that do not adhere to classical diffusive behavior; however, one major challenge in nonlocal simulations is the selection of model parameters.In this work we propose an optimization-based approach to parameter identification for fractional models with an optional truncation radius. We formulate the inference problem as an optimal control problem where the objective is to minimize the discrepancy between observed data and an approximate solution of the model, and the control variables are the fractional order and the truncation length. For the numerical solution of the minimization problem we propose a gradient-based approach, where we enhance the numerical performance by an approximation of the bilinear form of the state equation and its derivative with respect to the fractional order. Several numerical tests in one and two dimensions illustrate the theoretical results and show the robustness and applicability of our method.

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nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applications

Cited by 35 publications

References 32 publications

A general framework for substructuring-based domain decomposition methods for models having nonlocal interactions

A general framework for substructuring-based domain decomposition methods for models having nonlocal interactions

Physics-informed neural networks for solving parametric magnetostatic problems

An optimization-based approach to parameter learning for fractional type nonlocal models

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