DeepXDE: A deep learning library for solving differential equations

Abstract

Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation. The PINN algorithm is simple, and it can be applied to different types of PDEs, including integro-differential equations, fractional PDEs, and stochastic PDEs. Moreover, from the implementation point of view, PINNs solve inverse problems as easily as forward problems. We propose a new residual-based adaptive refinement (RAR) method to improve the training efficiency of PINNs. For pedagogical reasons, we compare the PINN algorithm to a standard finite element method. We also present a Python library for PINNs, DeepXDE, which is designed to serve both as an education tool to be used in the classroom as well as a research tool for solving problems in computational science and engineering. Specifically, DeepXDE can solve forward problems given initial and boundary conditions, as well as inverse problems given some extra measurements. DeepXDE supports complex-geometry domains based on the technique of constructive solid geometry, and enables the user code to be compact, resembling closely the mathematical formulation. We introduce the usage of DeepXDE and its customizability, and we also demonstrate the capability of PINNs and the user-friendliness of DeepXDE for five different examples. More broadly, DeepXDE contributes to the more rapid development of the emerging Scientific Machine Learning field.

References in corpus (18)

• Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks
• Neural Architecture Search with Reinforcement Learning
• DGM: A deep learning algorithm for solving partial differential equations
• Solving high-dimensional partial differential equations using deep learning
• Adaptive activation functions accelerate convergence in deep and physics-informed neural networks
• fPINNs: Fractional Physics-Informed Neural Networks
• Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data
• A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems
• A unified deep artificial neural network approach to partial differential equations in complex geometries
• Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks
• Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems
• Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport
• A Review of automatic differentiation and its efficient implementation
• Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations
• A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations
• On the Spectral Bias of Neural Networks
• Learning Parameters and Constitutive Relationships with Physics Informed Deep Neural Networks
• A Geometric Theory of Higher-Order Automatic Differentiation

Cited by in corpus (140)

• DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
• NSFnets (Navier-Stokes Flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations
• B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data
• 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
• PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain
• Differentiable modeling to unify machine learning and physical models and advance Geosciences
• A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications
• Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons
• Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport
• CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method
• Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations
• Physics-Informed Neural Network for Modelling the Thermochemical Curing Process of Composite-Tool Systems During Manufacture
• Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems
• Stiff-PINN: Physics-Informed Neural Network for Stiff Chemical Kinetics
• SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics
• PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs
• Uncovering near-wall blood flow from sparse data with physics-informed neural networks
• Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarios
• A Physics-Guided Neural Network Framework for Elastic Plates: Comparison of Governing Equations-Based and Energy-Based Approaches
• DeepM&Mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks
• Deep-learning-based surrogate flow modeling and geological parameterization for data assimilation in 3D subsurface flow
• Autonomous Discovery of Unknown Reaction Pathways from Data by Chemical Reaction Neural Network
• Multi-Objective Loss Balancing for Physics-Informed Deep Learning
• Physics Informed Neural Networks for Simulating Radiative Transfer
• Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training
• Physics-Informed Neural Operator for Learning Partial Differential Equations
• Hybrid thermal modeling of additive manufacturing processes using physics-informed neural networks for temperature prediction and parameter identification
• Physics-Informed Neural Network Method for Forward and Backward Advection-Dispersion Equations
• Meta-learning PINN loss functions
• Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism
• Reliable extrapolation of deep neural operators informed by physics or sparse observations
• Residual-based adaptivity for two-phase flow simulation in porous media using Physics-informed Neural Networks
• When and why PINNs fail to train: A neural tangent kernel perspective
• Characterizing possible failure modes in physics-informed neural networks
• Physics-informed neural networks of the Saint-Venant equations for downscaling a large-scale river model
• SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs
• Learning in Sinusoidal Spaces with Physics-Informed Neural Networks
• CENN: Conservative energy method based on neural networks with subdomains for solving variational problems involving heterogeneous and complex geometries
• An overview on deep learning-based approximation methods for partial differential equations
• 2022 Review of Data-Driven Plasma Science
• Thermodynamics and its Prediction and CALPHAD Modeling: Review, State of the Art, and Perspectives
• Solving inverse-PDE problems with physics-aware neural networks
• Solving forward and inverse problems of the logarithmic nonlinear Schrodinger equation with PT-symmetric harmonic potential via deep learning
• Physics-informed neural networks for imaging and parameter retrieval of photonic nanostructures from near-field data
• Magnetostatics and micromagnetics with physics informed neural networks
• Physics-informed neural networks for solving parametric magnetostatic problems
• Pre-training strategy for solving evolution equations based on physics-informed neural networks
• Residual-based physics-informed transfer learning: A hybrid method for accelerating long-term CFD simulations via deep learning
• Poisson CNN: Convolutional neural networks for the solution of the Poisson equation on a Cartesian mesh
• Algorithmically-Consistent Deep Learning Frameworks for Structural Topology Optimization
• A Deep Learning Approach for Predicting Two-dimensional Soil Consolidation Using Physics-Informed Neural Networks (PINN)
• Data-driven peakon and periodic peakon travelling wave solutions of some nonlinear dispersive equations via deep learning
• Conditional physics informed neural networks
• Choose a Transformer: Fourier or Galerkin
• Data-driven discovery of Green's functions with human-understandable deep learning
• Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems
• Physics-Informed Supervised Residual Learning for Electromagnetic Modeling
• Optimizing a DIscrete Loss (ODIL) to solve forward and inverse problems for partial differential equations using machine learning tools
• Applications of physics informed neural operators
• Improved generalization with deep neural operators for engineering systems: Path towards digital twin
• Embedding Hard Physical Constraints in Neural Network Coarse-Graining of 3D Turbulence
• A rate of convergence of Physics Informed Neural Networks for the linear second order elliptic PDEs
• Bayesian Physics-Informed Neural Networks for the Subsurface Tomography based on the Eikonal Equation
• On quadrature rules for solving Partial Differential Equations using Neural Networks
• Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEs
• ReLU Deep Neural Networks from the Hierarchical Basis Perspective
• Numerical methods for backward stochastic differential equations: A survey
• Differentiable Molecular Simulations for Control and Learning
• On Physics-Informed Neural Networks for Quantum Computers
• Physics-driven Learning of the Steady Navier-Stokes Equations using Deep Convolutional Neural Networks
• Efficient nonlinear manifold reduced order model
• Neural Eikonal Solver: improving accuracy of physics-informed neural networks for solving eikonal equation in case of caustics
• BubbleNet: Inferring micro-bubble dynamics with semi-physics-informed deep learning
• NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation framework
• Deep unfitted Nitsche method for elliptic interface problems
• Data-driven discoveries of Bäcklund transforms and soliton evolution equations via deep neural network learning schemes
• A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data
• Numerical Approximation of Partial Differential Equations by a Variable Projection Method with Artificial Neural Networks
• A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networks
• Neural Networks Enforcing Physical Symmetries in Nonlinear Dynamical Lattices: The Case Example of the Ablowitz-Ladik Model
• Data-driven Control of Agent-based Models: an Equation/Variable-free Machine Learning Approach
• DeepM&Mnet for hypersonics: Predicting the coupled flow and finite-rate chemistry behind a normal shock using neural-network approximation of operators
• Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks
• A Dimension-Augmented Physics-Informed Neural Network (DaPINN) with High Level Accuracy and Efficiency
• Modeling intermolecular and intramolecular modes of liquid water using multiple heat baths: Machine learning approach
• Data-driven Identification of 2D Partial Differential Equations using extracted physical features
• Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities
• A priori and a posteriori error estimates for the Deep Ritz method applied to the Laplace and Stokes problem
• An unsupervised learning approach to solving heat equations on chip based on Auto Encoder and Image Gradient
• Predicting nonlinear optical scattering with physics-driven neural networks
• RAR-PINN algorithm for the data-driven vector-soliton solutions and parameter discovery of coupled nonlinear equations
• Using physics-informed neural networks to compute quasinormal modes
• Neural network tokamak equilibria with incompressible flows
• Physics-informed neural networks for solving forward and inverse flow problems via the Boltzmann-BGK formulation
• L-HYDRA: Multi-Head Physics-Informed Neural Networks
• Machine learning based data-driven discovery of nonlinear phase-field dynamics
• Controlling Chaos in Van Der Pol Dynamics Using Signal-Encoded Deep Learning
• Deep Learning of Conjugate Mappings
• Towards fast weak adversarial training to solve high dimensional parabolic partial differential equations using XNODE-WAN
• Adapting Physics-Informed Neural Networks to Improve ODE Optimization in Mosquito Population Dynamics
• Data-driven discovery of multiscale chemical reactions governed by the law of mass action
• The Deep Minimizing Movement Scheme
• A Data-driven Multi-fidelity Physics-informed Learning Framework for Smart Manufacturing: A Composites Processing Case Study
• Physics-informed neural networks for learning the homogenized coefficients of multiscale elliptic equations
• Extreme learning machine collocation for the numerical solution of elliptic PDEs with sharp gradients
• A Priori Analysis of Stable Neural Network Solutions to Numerical PDEs
• Robust Modeling of Unknown Dynamical Systems via Ensemble Averaged Learning
• Certified machine learning: A posteriori error estimation for physics-informed neural networks
• Structure Preservation for the Deep Neural Network Multigrid Solver
• Bridging Fidelities to Predict Nanoindentation Tip Radii Using Interpretable Deep Learning Models
• Dynamics-informed deconvolutional neural networks for super-resolution identification of regime changes in epidemiological time series
• Generative Adversarial Networks with Physical Evaluators for Spray Simulation of Pintle Injector
• Deep FPF: Gain function approximation in high-dimensional setting
• Power Series Expansion Neural Network
• Optimal Transport for Parameter Identification of Chaotic Dynamics via Invariant Measures
• Automated differential equation solver based on the parametric approximation optimization
• Learning the Tangent Space of Dynamical Instabilities from Data
• SPADE4: Sparsity and Delay Embedding based Forecasting of Epidemics
• 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
• One-shot learning for solution operators of partial differential equations
• Living in the Physics and Machine Learning Interplay for Earth Observation
• Quantum Extremal Learning
• Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Space
• A Neural Solver for Variational Problems on CAD Geometries with Application to Electric Machine Simulation
• A Model-Constrained Tangent Slope Learning Approach for Dynamical Systems
• Data-driven method to learn the most probable transition pathway and stochastic differential equations
• Deeply Learned Spectral Total Variation Decomposition
• Uncertainty Quantification of Locally Nonlinear Dynamical Systems using Neural Networks
• Learning emergent PDEs in a learned emergent space
• Fixed-budget online adaptive learning for physics-informed neural networks. Towards parameterized problem inference
• Investigation of Numerical Diffusion in Aerodynamic Flow Simulations with Physics Informed Neural Networks
• Physics-informed neural networks for non-Newtonian fluid thermo-mechanical problems: an application to rubber calendering process
• Numerical Solution of Stiff ODEs with Physics-Informed RPNNs
• Physics-Informed Deep Learning Characterizes Morphodynamics of Asian Soybean Rust Disease
• A Taylor Based Sampling Scheme for Machine Learning in Computational Physics
• Traveling Wave Solutions of Partial Differential Equations via Neural Networks