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Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next
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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.
Forward citations
Cited by 8 Pith papers
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Unbiased Data-Driven Determination of the Nuclear Dipole Amplitude in the Color Glass Condensate
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A whole-log transformer with a geology-aware loss predicts stratigraphic zones and marker depths, beating sliding-window baselines on three well-log datasets.
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Physics-Informed Global Extraction of the Universal Small-$x$ Dipole Amplitude
A nonparametric PINN-constrained dipole amplitude simultaneously describes HERA total/charm DIS and J/psi photoproduction while maintaining Fourier positivity, unlike analytic MV-type fits.
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LIGO-PINN: Learned Initialization via Gated Optimization to Alleviate Convergence Failures in Physics Informed Neural Networks
Meta-learning on easy PDE tasks plus a layer-wise gating schedule reduces extrapolation error by about 91% relative to six PINN baselines on hard convection, Helmholtz, and Navier-Stokes benchmarks.
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Physics-Informed Residual Deep Learning for Constitutive Modeling of Hot Deformation and Dynamic Recrystallization in a Mo-Rich $\alpha+\beta$ Titanium Alloy
A DRX-aware physics-constrained neural network reproduces hot-compression flow stress of Ti–6Al–4Mo–1V–0.1Si with R²≈0.985, but its recrystallization fraction is a soft-prior artifact rather than an independently vali...
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Evaluation of Neural Surrogates for Physical Modelling Synthesis of Nonlinear Elastic Plates
On a Berger plate benchmark, state-of-the-art neural surrogates fail in long autoregressive rollouts, and time-domain error metrics miss the resulting spectral errors.
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Applications and Manipulations of Physics-Informed Neural Networks in Solving Differential Equations
The paper demonstrates standard PINN fitting for polynomial and heat-equation problems and claims PINNs are less sensitive than finite-difference methods to the CFL stability condition.
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