fPINNs: Fractional Physics-Informed Neural Networks

Abstract

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.

29 pages, 15 figures, 5 tables

References in corpus (8)

• Improving neural networks by preventing co-adaptation of feature detectors
• Neural Architecture Search with Reinforcement Learning
• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems
• Adaptive multi-fidelity polynomial chaos approach to Bayesian inference in inverse problems
• Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data
• Neural-net-induced Gaussian process regression for function approximation and PDE solution
• Recovering an Unknown Source in a Fractional Diffusion Problem

Cited by in corpus (69)

• DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
• DeepXDE: A deep learning library for solving differential equations
• hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition
• Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems
• A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks
• PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs
• Self-adaptive loss balanced Physics-informed neural networks for the incompressible Navier-Stokes equations
• Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons
• Deep Autoencoder based Energy Method for the Bending, Vibration, and Buckling Analysis of Kirchhoff Plates
• Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations
• DeepM&Mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks
• Numerical methods for nonlocal and fractional models
• A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder
• Non-invasive Inference of Thrombus Material Properties with Physics-informed Neural Networks
• A two-stage physics-informed neural network method based on conserved quantities and applications in localized wave solutions
• Physics Informed Neural Networks for Simulating Radiative Transfer
• Physics-informed PointNet: A deep learning solver for steady-state incompressible flows and thermal fields on multiple sets of irregular geometries
• A Review of Machine Learning Methods Applied to Structural Dynamics and Vibroacoustic
• Physics-Informed Neural Network Method for Forward and Backward Advection-Dispersion Equations
• Residual-based adaptivity for two-phase flow simulation in porous media using Physics-informed Neural Networks
• Deep learning of free boundary and Stefan problems
• Distributed physics informed neural network for data-efficient solution to partial differential equations
• Temperature Field Inversion of Heat-Source Systems via Physics-Informed Neural Networks
• Fractional SEIR Model and Data-Driven Predictions of COVID-19 Dynamics of Omicron Variant
• Physics-informed neural networks for imaging and parameter retrieval of photonic nanostructures from near-field data
• Data-driven learning of robust nonlocal physics from high-fidelity synthetic data
• On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton--Jacobi partial differential equations
• Physics-informed neural network methods based on Miura transformations and discovery of new localized wave solutions
• Weak Form Theory-guided Neural Network (TgNN-wf) for Deep Learning of Subsurface Single and Two-phase Flow
• Surrogate and inverse modeling for two-phase flow in porous media via theory-guided convolutional neural network
• Pre-training strategy for solving evolution equations based on physics-informed neural networks
• Physics-informed neural networks method in high-dimensional integrable systems
• Data-driven peakon and periodic peakon travelling wave solutions of some nonlinear dispersive equations via deep learning
• DySMHO: Data-Driven Discovery of Governing Equations for Dynamical Systems via Moving Horizon Optimization
• Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems
• Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domains
• Convergence Rate Analysis for Deep Ritz Method
• Adaptive deep density approximation for Fokker-Planck equations
• Applications of physics informed neural operators
• Deep learning neural networks for the third-order nonlinear Schrodinger equation: Solitons, breathers, and rogue waves
• A rate of convergence of Physics Informed Neural Networks for the linear second order elliptic PDEs
• Deep learning for thermal plasma simulation: solving 1-D arc model as an example
• Recipes for when Physics Fails: Recovering Robust Learning of Physics Informed Neural Networks
• On quadrature rules for solving Partial Differential Equations using Neural Networks
• Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networks
• Error estimates of residual minimization using neural networks for linear PDEs
• Energy-based error bound of physics-informed neural network solutions in elasticity
• On Physics-Informed Neural Networks for Quantum Computers
• The Neural Network Approach to Inverse Problems in Differential Equations
• Bilevel parameter learning for nonlocal image denoising models
• Fully probabilistic deep models for forward and inverse problems in parametric PDEs
• Data-Driven Discovery of Coarse-Grained Equations
• Data-driven discoveries of Bäcklund transforms and soliton evolution equations via deep neural network learning schemes
• Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs
• A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data
• Neural network architectures using min-plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEs
• PCNN: A physics-constrained neural network for multiphase flows
• SVD-PINNs: Transfer Learning of Physics-Informed Neural Networks via Singular Value Decomposition
• Solving Non-local Fokker-Planck Equations by Deep Learning
• Efficient quadrature rules for finite element discretizations of nonlocal equations
• Overcoming the curse of dimensionality for some Hamilton--Jacobi partial differential equations via neural network architectures
• CDNNs: The coupled deep neural networks for coupling of the Stokes and Darcy-Forchheimer problems
• Learning Green's Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver
• Data-Centric Engineering: integrating simulation, machine learning and statistics. Challenges and Opportunities
• Sinc Kolmogorov-Arnold network and its application for solving PDEs with singularities
• Fractional models of Reynolds-averaged Navier-Stokes equations for Turbulent flows
• Stochastic spectral embedding
• Learning To Estimate Regions Of Attraction Of Autonomous Dynamical Systems Using Physics-Informed Neural Networks
• Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs