CAN-PINN: A fast physics-informed neural network based on coupled-automatic–numerical differentiation method

Chiu, Pao‐Hsiung; Wong, Jia Haur; Ooi, Chin Chun; Dao, My Ha; Ong, Yew-Soon

doi:10.1016/j.cma.2022.114909

Cited by 124 publications

(33 citation statements)

References 49 publications

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“…Further research can look into alternative or hybrid methods of differentiating the differential equations. To speed up PINN training, the loss function in Chiu et al [33] is defined using numerical differentiation and automatic differentiation. The proposed can-PINN, i.e.…”

Section: Improving Implementation Aspects In Pinnmentioning

confidence: 99%

Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next

et al. 2022

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Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, integral-differential equations, and stochastic PDEs. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related advantages and disadvantages. The review also attempts to incorporate publications on a broader range of collocation-based physics informed neural networks, which stars form the vanilla PINN, as well as many other variants, such as physics-constrained neural networks (PCNN), variational hp-VPINN, and conservative PINN (CPINN). The study indicates that most research has focused on customizing the PINN through different activation functions, gradient optimization techniques, neural network structures, and loss function structures. Despite the wide range of applications for which PINNs have been used, by demonstrating their ability to be more feasible in some contexts than classical numerical techniques like Finite Element Method (FEM), advancements are still possible, most notably theoretical issues that remain unresolved.

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Section: Improving Implementation Aspects In Pinnmentioning

confidence: 99%

Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next

et al. 2022

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“…Wide varieties of PINNs have been found in the literature in recent times. can-PINNs [15] link derivative terms with nearby support points, which generally apply to Taylor series expansion-based numerical systems. Apart from demonstrating good dispersion and dissipation characteristics, they are highly trainable and require four to sixteen times fewer collocation points than original PINNs.…”

Section: Literature Review a Physics Informed Neural Networkmentioning

confidence: 99%

Physics-informed Neural Networks approach to solve the Blasius function

Greeshma¹,

Nair²,

Nair³

et al. 2023

Preprint

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Deep learning techniques with neural networks have been used effectively in computational fluid dynamics (CFD) to obtain solutions to nonlinear differential equations. This paper presents a physics-informed neural network (PINN) approach to solve the Blasius function. This method eliminates the process of changing the non-linear differential equation to an initial value problem. Also, it tackles the convergence issue arising in the conventional series solution. It is seen that this method produces results that are at par with the numerical and conventional methods. The solution is extended to the negative axis to show that PINNs capture the singularity of the function at η = −5.69.

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“…While both methods have their pros and cons, ND-loss can be flexibly implemented across many different neural network (NN) architectures, including both multi-layer perceptrons (MLPs) and convolutional neural networks (CNNs), because they do not require the NN to retain differentiability, unlike AD. Recent studies [4]- [6] have suggested that ND-type methods and especially coupledautomatic-numerical differentiation (CAN)-loss [6] can more robustly and efficiently produce accurate solutions with fewer training samples, whereas conventional AD-loss is prone to failure during training. This is because ND-type methods approximate high order derivatives using PINN output from neighbouring samples, hence, they can effectively connect sparse samples into piecewise regions via these local approximations, thereby facilitating fast physics-informed learning across the entire domain with sparser samples.…”

Section: Introductionmentioning

confidence: 99%

LSA-PINN: Linear Boundary Connectivity Loss for Solving PDEs on Complex Geometry

Wong¹,

Chiu²,

Ooi³

et al. 2023

Preprint

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We present a novel loss formulation for efficient learning of complex dynamics from governing physics, typically described by partial differential equations (PDEs), using physicsinformed neural networks (PINNs). In our experiments, existing versions of PINNs are seen to learn poorly in many problems, especially for complex geometries, as it becomes increasingly difficult to establish appropriate sampling strategy at the near boundary region. Overly dense sampling can adversely impede training convergence if the local gradient behaviors are too complex to be adequately modelled by PINNs. On the other hand, if the samples are too sparse, existing PINNs tend to overfit the near boundary region, leading to incorrect solution. To prevent such issues, we propose a new Boundary Connectivity (BCXN) loss function which provides linear local structure approximation (LSA) to the gradient behaviors at the boundary for PINN. Our BCXN-loss implicitly imposes local structure during training, thus facilitating fast physics-informed learning across entire problem domains with order of magnitude sparser training samples. This LSA-PINN method shows a few orders of magnitude smaller errors than existing methods in terms of the standard L2norm metric, while using dramatically fewer training samples and iterations. Our proposed LSA-PINN does not pose any requirement on the differentiable property of the networks, and we demonstrate its benefits and ease of implementation on both multi-layer perceptron and convolutional neural network versions as commonly used in current PINN literature.

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CAN-PINN: A fast physics-informed neural network based on coupled-automatic–numerical differentiation method

Cited by 124 publications

References 49 publications

Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next

Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next

Physics-informed Neural Networks approach to solve the Blasius function

LSA-PINN: Linear Boundary Connectivity Loss for Solving PDEs on Complex Geometry

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