AL-PINNs: Augmented Lagrangian relaxation method for Physics-Informed Neural Networks

Son, Hwijae; Cho, Sung Woong; Hwang, Hyung Ju

doi:10.48550/arxiv.2205.01059

Cited by 1 publication

(2 citation statements)

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“…It should be mentioned that setting γ = 1 led to failure in almost all of the synthetic and experimental implementations of PINNs in this study. Gauging of loss function weights has been the subject of extensive recent studies [12,25,47,26,27,28]. One systematic approach is the adaptive SA-PINNs [12] where the multiplier γ(ξ, ω; τ ) is a distributed parameter of û whose value is updated in each epoch according to a minimax weighting paradigm.…”

Section: Physics-informed Neural Networkmentioning

confidence: 99%

“…PINNs have recently come under the spotlight for offering efficient, yet predictive, models of complex PDE systems [10] that has so far been backed by rigorous theoretical justification within the context of linear elliptic and parabolic PDEs [23]. Given the multitask nature of training for these networks and the existing challenges with modeling stiff and highly oscillatory PDEs [12,24], much of the most recent efforts has been focused on (a) adaptive gauging of the loss function [12,25,26,27,28,29,13], and (b) addressing the gradient pathologies [24,13] e.g., via learning rate annealing [30] and customizing the network architecture [11,31,32]. In this study, our initially austere implementations of PINNs using both synthetic and experimental waveforms led almost invariably to failure which further investigation attributed to the following impediments: (a) highnorm gradient fields due to large wavenumbers, (b) high-order governing PDEs in the case of laboratory experiments, and (c) imbalanced objectives in the loss function.…”

Section: Introductionmentioning

confidence: 99%

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Deep learning for full-field ultrasonic characterization

Xu¹,

Pourahmadian²,

Song³

et al. 2023

Preprint

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This study takes advantage of recent advances in machine learning to establish a physics-based data analytic platform for distributed reconstruction of mechanical properties in layered components from full waveform data. In this vein, two logics, namely the direct inversion and physics-informed neural networks (PINNs), are explored. The direct inversion entails three steps: (i) spectral denoising and differentiation of the full-field data, (ii) building appropriate neural maps to approximate the profile of unknown physical and regularization parameters on their respective domains, and (iii) simultaneous training of the neural networks by minimizing the Tikhonov-regularized PDE loss using data from (i). PINNs furnish efficient surrogate models of complex systems with predictive capabilities via multitask learning where the field variables are modeled by neural maps endowed with (scaler or distributed) auxiliary parameters such as physical unknowns and loss function weights. PINNs are then trained by minimizing a measure of data misfit subject to the underlying physical laws as constraints. In this study, to facilitate learning from ultrasonic data, the PINNs loss adopts (a) wavenumber-dependent Sobolev norms to compute the data misfit, and (b) non-adaptive weights in a specific scaling framework to naturally balance the loss objectives by leveraging the form of PDEs germane to elasticwave propagation. Both paradigms are examined via synthetic and laboratory test data. In the latter case, the reconstructions are performed at multiple frequencies and the results are verified by a set of complementary experiments highlighting the importance of verification and validation in data-driven modeling.

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