Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations

Abstract

We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential equations. In this second part of our two-part treatise, we focus on the problem of data-driven discovery of partial differential equations. Depending on whether the available data is scattered in space-time or arranged in fixed temporal snapshots, we introduce two main classes of algorithms, namely continuous time and discrete time models. The effectiveness of our approach is demonstrated using a wide range of benchmark problems in mathematical physics, including conservation laws, incompressible fluid flow, and the propagation of nonlinear shallow-water waves.

Cited by in corpus (60)

• Explainable Machine Learning for Scientific Insights and Discoveries
• Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks
• Integrating Machine Learning and Multiscale Modeling: Perspectives, Challenges, and Opportunities in the Biological, Biomedical, and Behavioral Sciences
• PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network
• Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems
• A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate
• A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications
• The Challenge of Machine Learning in Space Weather Nowcasting and Forecasting
• Machine Learning in Aerodynamic Shape Optimization
• DeepM&Mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks
• Deep learning of dynamics and signal-noise decomposition with time-stepping constraints
• Data-Driven Deep Learning of Partial Differential Equations in Modal Space
• Multi-Objective Loss Balancing for Physics-Informed Deep Learning
• Simulator-free Solution of High-Dimensional Stochastic Elliptic Partial Differential Equations using Deep Neural Networks
• Data-driven discovery of PDEs in complex datasets
• Solving Fokker-Planck equation using deep learning
• Probabilistic neural networks for fluid flow surrogate modeling and data recovery
• D3M: A deep domain decomposition method for partial differential equations
• DeepMoD: Deep learning for Model Discovery in noisy data
• Applications of deep learning in traffic congestion detection, prediction and alleviation: A survey
• Time-series learning of latent-space dynamics for reduced-order model closure
• Physics-informed linear regression is competitive with two Machine Learning methods in residential building MPC
• An overview on deep learning-based approximation methods for partial differential equations
• Explore missing flow dynamics by physics-informed deep learning: the parameterised governing systems
• Solving inverse-PDE problems with physics-aware neural networks
• Synchronization to big-data: nudging the Navier-Stokes equations for data assimilation of turbulent flows
• On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton--Jacobi partial differential equations
• Data driven approximation of parametrized PDEs by Reduced Basis and Neural Networks
• Latent Space Data Assimilation by using Deep Learning
• Active Training of Physics-Informed Neural Networks to Aggregate and Interpolate Parametric Solutions to the Navier-Stokes Equations
• Physics-informed data based neural networks for two-dimensional turbulence
• PI-VAE: Physics-Informed Variational Auto-Encoder for stochastic differential equations
• Efficient Deep Learning Techniques for Multiphase Flow Simulation in Heterogeneous Porous Media
• On generalized residue network for deep learning of unknown dynamical systems
• Adaptive deep density approximation for Fokker-Planck equations
• A Physics-Informed Neural Network approach for compartmental epidemiological models
• Deep Learning Ensemble for Real-time Gravitational Wave Detection of Spinning Binary Black Hole Mergers
• Applications of physics informed neural operators
• Neural Network Forecast of the Sunspot Butterfly Diagram
• A physics-aware, probabilistic machine learning framework for coarse-graining high-dimensional systems in the Small Data regime
• What machine learning can do for computational solid mechanics
• MINN: Learning the dynamics of differential-algebraic equations and application to battery modeling
• BubbleNet: Inferring micro-bubble dynamics with semi-physics-informed deep learning
• Physics-Informed Machine Learning for Grade Prediction in Froth Flotation
• Magnetohydrodynamics with Physics Informed Neural Operators
• Data-driven, multi-moment fluid modeling of Landau damping
• Deep Learning of Conjugate Mappings
• Physics-Informed Neural Networks for Securing Water Distribution Systems
• Towards Model Reduction for Power System Transients with Physics-Informed PDE
• Bridging the Gap: Machine Learning to Resolve Improperly Modeled Dynamics
• Bayesian Sparse learning with preconditioned stochastic gradient MCMC and its applications
• SubTSBR to tackle high noise and outliers for data-driven discovery of differential equations
• Physics-Informed Neural Networks for Transonic Flows around an Airfoil
• Analytical derivatives of Neural Networks
• Neural Network Analysis of S-Star Dynamics: Implications for Modified Gravity
• Residual U-Net for accurate and efficient prediction of hemodynamics in two-dimensional asymmetric stenosis
• Neural Stream Functions
• Discovery of Quasi-Integrable Equations from traveling-wave data using the Physics-Informed Neural Networks
• Parameter Estimation and Adaptive Solution of the Leray-Burgers Equation Using Physics-Informed Neural Networks
• MFPC-Net: Multi-fidelity Physics-Constrained Neural Process