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From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning
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Physics-Informed Neural Networks (PINNs) have emerged as a key tool in Scientific Machine Learning since their introduction in 2017, enabling the efficient solution of ordinary and partial differential equations using sparse measurements. Over the past few years, significant advancements have been made in the training and optimization of PINNs, covering aspects such as network architectures, adaptive refinement, domain decomposition, and the use of adaptive weights and activation functions. A notable recent development is the Physics-Informed Kolmogorov-Arnold Networks (PIKANS), which leverage a representation model originally proposed by Kolmogorov in 1957, offering a promising alternative to traditional PINNs. In this review, we provide a comprehensive overview of the latest advancements in PINNs, focusing on improvements in network design, feature expansion, optimization techniques, uncertainty quantification, and theoretical insights. We also survey key applications across a range of fields, including biomedicine, fluid and solid mechanics, geophysics, dynamical systems, heat transfer, chemical engineering, and beyond. Finally, we review computational frameworks and software tools developed by both academia and industry to support PINN research and applications.
Forward citations
Cited by 5 Pith papers
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Cosmo-SPINN: Fuzzy Dark Matter Simulations with Physics-Informed Generative Networks
Physics-informed generative U-Nets evolve and super-resolve fuzzy dark matter fields under Schrödinger–Poisson constraints with far less supervised data than pure data-driven baselines.
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A physics-informed neural network approach to the point defect model for electrochemical oxide film growth
A hybrid PINN anchored by one FEM data point reproduces point-defect-model film thicknesses to about 1% error, while the pure PINN overpredicts by 2,400-5,700%.
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Leveraging KANs for Expedient Training of Multichannel MLPs via Preconditioning and Geometric Refinement
Training in a B-spline KAN basis is equivalent to preconditioned gradient descent on a multichannel ReLU MLP, and geometric refinement plus trainable knots accelerate and improve training.
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Multiprecision computing for multistage fractional physics-informed neural networks
A two-stage, multi-scale fPINN is claimed to reach 10^-7 accuracy, but the reported numbers are inconsistent with the L1 discretization error on the coarse grid.
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A Unified Framework for Simultaneous Parameter and Function Discovery in Differential Equations
The paper proves identifiability conditions for ODE inverse problems with one unknown constant and one unknown function, and adds approximate error bounds when data points are close but not identical.
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