Physics-informed neural networks for solving nonlinear Bloch equations in atomic magnetometry

Lei, G.; Ma, Ning; Sun, Bowen; Mao, Ke; Chen, Baodong; Zhai, Yueyang

doi:10.1088/1402-4896/ace290

Cited by 2 publications

(2 citation statements)

References 42 publications

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“…Forward and inverse problems found in differential equations can both be solved using PINNs [8][9][10]. PINNs are also efficient in solving non-linear partial differential equations which are sometimes challenging to handle using traditional methods [11][12][13]. It is observed that specific issues associated with dimensionality in the area of numerical analysis of differential and dynamical systems are resolved with the help of PINNs [14,15].…”

Section: Introductionmentioning

confidence: 99%

Gradient-based adaptive neural network technique for two-dimensional local fractional elliptic PDEs

Jha,

Mallik

2024

Phys. Scr.

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This paper introduces gradient-based adaptive neural networks to solve local fractional elliptic partial differential equations. The impact of physics-informed neural networks helps to approximate elliptic partial differential equations governed by the physical process. The proposed technique employs learning the behaviour of complex systems based on input-output data, and automatic differentiation ensures accurate computation of gradient. The method computes the singularity-embedded local fractional partial derivative model on a Hausdorff metric, which otherwise halts the computation by available approximating numerical methods. This is possible because the new network is capable of updating the weight associated with loss terms depending on the solution domain and requirement of solution behaviour. The semi-positive definite character of the neural tangent kernel achieves the convergence of gradient-based adaptive neural networks. The importance of hyperparameters, namely the number of neurons and the learning rate, is shown by considering a stationary anomalous diffusion-convection model on a rectangular domain. The proposed method showcases the network’s ability to approximate solutions of various local fractional elliptic partial differential equations with varying fractal parameters.

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Section: Introductionmentioning

confidence: 99%

Gradient-based adaptive neural network technique for two-dimensional local fractional elliptic PDEs

Jha,

Mallik

2024

Phys. Scr.

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“…A notable development is the introduction of automatic differentiation [1], which makes it possible to bypass numerical discretization and mesh generation, leading to a new machine learning-based technique known as PINN [2,3]. Since the introduction of the PINN, it has been widely used to approximate solution to PDE [4][5][6][7][8][9][10][11][12][13][14][15][16], particularly highly nonlinear ones with non-convex and oscillating behavior, such as Navier-Stokes equations, that pose challenges for traditional numerical discretization techniques. The problematic part of Navier-Stokes equations is because of the convective term in the equations introduces a nonlinearity due to the product of velocity components.…”

Section: Introductionmentioning

confidence: 99%

A novel discretized physics-informed neural network model applied to the Navier–Stokes equations

Khademi,

Dufour

2024

Phys. Scr.

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The advancement of scientific machine learning (ML) techniques has led to the development of methods for approximating solutions to nonlinear partial differential equations (PDE) with increased efficiency and accuracy. Automatic differentiation has played a pivotal role in this progress, enabling the creation of physics-informed neural networks (PINN) that integrate relevant physics into machine learning models. PINN have shown promise in approximating the solutions to the Navier-Stokes equations, overcoming limitations of traditional numerical discretization methods. However, challenges such as local minima and long training times persist, motivating the exploration of domain decomposition techniques to improve it. Previous domain decomposition models have introduced spatial and temporal domain decompositions but have yet to fully address issues of smoothness and regularity of global solution. In this study, we present a novel domain decomposition approach for PINN, termed domain discretized PINN (DD-PINN), which incorporates complementary loss functions, subdomain-specific transformer networks (TRF), and independent optimization within each subdomain. By enforcing continuity and differentiability through interface constraints and leveraging the Sobolev (H1) norm of the mean squared error (MSE), rather than the Euclidean norm (L2), DD-PINN enhances solution regularity and accuracy. The inclusion of TRF in each subdomain facilitates feature extraction and improves convergence rates, as demonstrated through simulations of two test problems: steady-state flow in a two-dimensional lid-driven cavity and the time-dependent cylinder wake. Numerical comparisons highlight the effectiveness of DD-PINN in preserving global solution regularity and accurately approximating complex phenomena, marking a significant advancement over previous domain decomposition methods within the PINN framework.

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Physics-informed neural networks for solving nonlinear Bloch equations in atomic magnetometry

Cited by 2 publications

References 42 publications

Gradient-based adaptive neural network technique for two-dimensional local fractional elliptic PDEs

Gradient-based adaptive neural network technique for two-dimensional local fractional elliptic PDEs

A novel discretized physics-informed neural network model applied to the Navier–Stokes equations

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