fPINNs: Fractional Physics-Informed Neural Networks

By Guofei Pang, Lu Lu, G. Em Karniadakis. 2018

975 citations recorded by the source corpus.

Physics-informed neural networks (PINNs) are effective in solving integer-order partial differential equations (PDEs) based on scattered and noisy data. PINNs employ standard feedforward neural networks (NNs) with the PDEs explicitly encoded into the NN using automatic differentiation, while the sum of the mean-squared PDE-residuals and the mean-squared error in initial/boundary conditions is minimized with respect to the NN parameters. We extend PINNs to fractional PINNs (fPINNs) to solve space-time fractional advection-diffusion equations (fractional ADEs), and we demonstrate their accuracy and effectiveness in solving multi-dimensional forward and inverse problems with forcing terms whose values are only known at randomly scattered spatio-temporal coordinates (black-box forcing terms). A novel element of the fPINNs is the hybrid approach that we introduce for constructing the residual in the loss function using both automatic differentiation for the integer-order operators and numerical discretization for the fractional operators. We consider 1D time-dependent fractional ADEs and compare white-box (WB) and black-box (BB) forcing. We observe that for the BB forcing fPINNs outperform FDM. Subsequently, we consider multi-dimensional time-, space-, and space-time-fractional ADEs using the directional fractional Laplacian and we observe relative errors of $10^{-4}$. Finally, we solve several inverse problems in 1D, 2D, and 3D to identify the fractional orders, diffusion coefficients, and transport velocities and obtain accurate results even in the presence of significant noise.

Subfields: Biomedical Engineering, Modeling and Simulation, Statistical and Nonlinear Physics, Model Architectures, AI for Physical Sciences

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Citations in the corpus

Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges (2021, methodology): “PINNs have been extended to convolutional (Zhu et al., ∂x2 with initial condition u(x, 0) = f(x) and Dirichlet boundary conditions u(0, t) = 0 and u(L, t) = 0.”
PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs (2019, background): “Examples include the Navier-Stokes and the KdV equations [1], stochastic PDEs [13–16], and fractional PDEs [17].”
DeepXDE: A Deep Learning Library for Solving Differential Equations (2019, methodology): “Finally we note that with AD we differentiate the NN and therefore we can deal with noisy data [42].”
Physics-informed neural networks (PINNs) for fluid mechanics: a review (2021, methodology): “Beneﬁcial due to the high ﬂexibility and the expressive ability in function approximation, PINNs have been extended to solve various classes of PDEs, e.g., integro-differential equations [21], fractional equations [21], surfaces PDEs [22] and stochastic differential equations [23].”
A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks (2022, methodology): “In addition to standard PDEs, PINNs have also been extended to solve other types of PDEs, including integro-diﬀerential equations [3], fractional PDEs [9], and stochastic PDEs [10].”
Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs (2020, background): “There has been an explosive growth of papers that present PINNs for different applications and a very incomplete list of references includes [15,27,29,39,44,51,54,61].”
Variational Physics-Informed Neural Networks For Solving Partial Differential Equations (2019, background): “The recent development of PINNs has been well established for forward and inverse problem of solving di ﬀ erential equations [30, 32, 39, 29] and since then a number of extensions has been made to tackle several physical and biomedical problems [28, 18, 19].”
A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data (2021, methodology): “For example, using the DeepONet as a building block in [43] for modeling hypersonic ﬂows, we were able to obtain very accurate results outside the Mach number training range [8 , 10], e.g., at Mach number 7 and 11, by assimilating only up to ﬁve new measurements of velocity and temperature.”
hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition (2020, background): “…employed in many physical problems such as discovering turbulence models from scattered / noisy measurements [28], high speed ﬂows [4], stochastic di ﬀ erential equation by generative adversarial networks [24], fractional di ﬀ erential equations [29], and adaptive activation functions [30, 31].”
Analyses of internal structures and defects in materials using physics-informed neural networks (2022, background): “Different residual terms (PDEs, BCs, and data) require different residual points.”
Physics-informed neural networks with hard constraints for inverse design (2021, methodology): “Moreover, PINNs have been successfully employed to solve diverse inverse problems , including in optics [8], ﬂuid mechanics [45, 55], systems biology [60], biomedicine [46, 24], and even inverse problems of stochastic PDEs [62] and fractional PDEs [38].”
When and why PINNs fail to train: A neural tangent kernel perspective (2020, background): “differential equations [19], fractional differential equations [20, 21], and beyond [22, 23, 24, 25].”
Extended Physics-informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition based Deep Learning Framework for Nonlinear Partial Differential Equations (2020, background): “Thanks to their universal approximation properties, they can be exploited to construct alternative approaches for solving PDEs.”
Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations (2021, methodology): “PINNs have been extended to fractional differential equations [Pang et al., 2019], stochastic differential equations [Yang et al., 2020, Zhang et al., 2019], and have even been used to discover the underlying differential equations themselves [Chen et al., 2020a].”
Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next (2022, methodology): “The solution proposed in Pang et al [128], is then extended in Kharazmi et al [77] where they address also time-dependent fractional orders.”
Self-adaptive loss balanced Physics-informed neural networks (2021, methodology): “These discretization-based methods approximate the solution of PDEs through their values at a set of grid points distributed over the spatial and temporal domain.”
Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems (2021, methodology): “In some PDEs, it is possible to enforce the boundary conditions exactly and automatically by modifying the network architecture [20, 4, 25, 8], which eliminates the loss term of boundary conditions.”