Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems

DOI10.1016/j.jcp.2019.07.048zbMath1454.65008arXiv1809.08327OpenAlexW2890968382MaRDI QIDQ2222519

Lu Lu, Dongkun Zhang, Ling Guo, George Em. Karniadakis

Publication date: 27 January 2021

Published in: Journal of Computational Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1809.08327

zbMATH Keywords

stochastic differential equationsuncertainty quantificationdropoutarbitrary polynomial chaosphysics-informed neural networks

Mathematics Subject Classification ID

Artificial neural networks and deep learning (68T07) Reaction-diffusion equations (35K57) Inverse problems for PDEs (35R30) Computational methods for stochastic equations (aspects of stochastic analysis) (60H35) Numerical solutions to stochastic differential and integral equations (65C30)

Related Items (92)

Solving forward and inverse problems of the logarithmic nonlinear Schrödinger equation with \(\mathcal{PT}\)-symmetric harmonic potential via deep learning ⋮ On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton-Jacobi partial differential equations ⋮ B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data ⋮ Active training of physics-informed neural networks to aggregate and interpolate parametric solutions to the Navier-Stokes equations ⋮ Physics-informed semantic inpainting: application to geostatistical modeling ⋮ Eigenfunction martingale estimating functions and filtered data for drift estimation of discretely observed multiscale diffusions ⋮ PhyGeoNet: physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain ⋮ Bayesian sparse learning with preconditioned stochastic gradient MCMC and its applications ⋮ Image inversion and uncertainty quantification for constitutive laws of pattern formation ⋮ DeepM\&Mnet: inferring the electroconvection multiphysics fields based on operator approximation by neural networks ⋮ Mutual information for explainable deep learning of multiscale systems ⋮ Theory-guided hard constraint projection (HCP): a knowledge-based data-driven scientific machine learning method ⋮ MIM: a deep mixed residual method for solving high-order partial differential equations ⋮ Clustered active-subspace based local Gaussian process emulator for high-dimensional and complex computer models ⋮ The mixed deep energy method for resolving concentration features in finite strain hyperelasticity ⋮ A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder ⋮ Multifidelity modeling for physics-informed neural networks (PINNs) ⋮ MIONet: Learning Multiple-Input Operators via Tensor Product ⋮ Thermodynamically consistent physics-informed neural networks for hyperbolic systems ⋮ Normalizing field flows: solving forward and inverse stochastic differential equations using physics-informed flow models ⋮ A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data ⋮ Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems ⋮ Learning generative neural networks with physics knowledge ⋮ A Neural Network Approach to Sampling Based Learning Control for Quantum System with Uncertainty ⋮ Scientific machine learning through physics-informed neural networks: where we are and what's next ⋮ A Symplectic Based Neural Network Algorithm for Quantum Controls under Uncertainty ⋮ Data-driven rogue waves and parameters discovery in nearly integrable \(\mathcal{PT}\)-symmetric Gross-Pitaevskii equations via PINNs deep learning ⋮ Stochastic physics-informed neural ordinary differential equations ⋮ Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data ⋮ A-PINN: auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations ⋮ Wasserstein generative adversarial uncertainty quantification in physics-informed neural networks ⋮ A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problems ⋮ Improved deep neural networks with domain decomposition in solving partial differential equations ⋮ A neural network-based approach for bending analysis of strain gradient nanoplates ⋮ Deep learning neural networks for the third-order nonlinear Schrödinger equation: bright solitons, breathers, and rogue waves ⋮ One-dimensional ice shelf hardness inversion: clustering behavior and collocation resampling in physics-informed neural networks ⋮ Data-driven method to learn the most probable transition pathway and stochastic differential equation ⋮ Data driven modeling of interfacial traction-separation relations using a thermodynamically consistent neural network ⋮ Deep learning methods for partial differential equations and related parameter identification problems ⋮ PI-VAE: physics-informed variational auto-encoder for stochastic differential equations ⋮ A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks ⋮ A dimension-augmented physics-informed neural network (DaPINN) with high level accuracy and efficiency ⋮ Uncertainty quantification in scientific machine learning: methods, metrics, and comparisons ⋮ A metalearning approach for physics-informed neural networks (PINNs): application to parameterized PDEs ⋮ A decoupled physics-informed neural network for recovering a space-dependent force function in the wave equation from integral overdetermination data ⋮ Neural network architectures using min-plus algebra for solving certain high-dimensional optimal control problems and Hamilton-Jacobi PDEs ⋮ Loss-attentional physics-informed neural networks ⋮ A new method for solving nonlinear partial differential equations based on liquid time-constant networks ⋮ Variational inference in neural functional prior using normalizing flows: application to differential equation and operator learning problems ⋮ A Normalizing Field Flow Induced Two-Stage Stochastic Homogenization Method for Random Composite Materials ⋮ Learning Specialized Activation Functions for Physics-Informed Neural Networks ⋮ Bi-Orthogonal fPINN: A Physics-Informed Neural Network Method for Solving Time-Dependent Stochastic Fractional PDEs ⋮ Label-free learning of elliptic partial differential equation solvers with generalizability across boundary value problems ⋮ Physics-informed information field theory for modeling physical systems with uncertainty quantification ⋮ NeuralUQ: A Comprehensive Library for Uncertainty Quantification in Neural Differential Equations and Operators ⋮ Bayesian Deep Learning Framework for Uncertainty Quantification in Stochastic Partial Differential Equations ⋮ Physics-informed variational inference for uncertainty quantification of stochastic differential equations ⋮ Bi-fidelity modeling of uncertain and partially unknown systems using DeepONets ⋮ Deep learning soliton dynamics and complex potentials recognition for 1D and 2D \(\mathcal{PT}\)-symmetric saturable nonlinear Schrödinger equations ⋮ Posteriori error neural network: a recovery type posteriori error estimator based on neural network for diffusion problems ⋮ Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations ⋮ An artificial neural network approach to identify the parameter in a nonlinear subdiffusion model ⋮ Variationally mimetic operator networks ⋮ A nonlinear-manifold reduced-order model and operator learning for partial differential equations with sharp solution gradients ⋮ Physics-constrained data-driven variational method for discrepancy modeling ⋮ Gradient-enhanced physics-informed neural networks based on transfer learning for inverse problems of the variable coefficient differential equations ⋮ An Adaptive Physics-Informed Neural Network with Two-Stage Learning Strategy to Solve Partial Differential Equations ⋮ Enhancing phenomenological yield functions with data: challenges and opportunities ⋮ Topology optimization based on deep representation learning (DRL) for compliance and stress-constrained design ⋮ Control of partial differential equations via physics-informed neural networks ⋮ An overview on deep learning-based approximation methods for partial differential equations ⋮ A composite neural network that learns from multi-fidelity data: application to function approximation and inverse PDE problems ⋮ DeepXDE: A Deep Learning Library for Solving Differential Equations ⋮ Physics-informed neural network for modelling the thermochemical curing process of composite-tool systems during manufacture ⋮ Bayesian neural networks for uncertainty quantification in data-driven materials modeling ⋮ Learning on dynamic statistical manifolds ⋮ Physics-Informed Neural Networks with Hard Constraints for Inverse Design ⋮ PPINN: parareal physics-informed neural network for time-dependent PDEs ⋮ Non-invasive inference of thrombus material properties with physics-informed neural networks ⋮ Deep autoencoder based energy method for the bending, vibration, and buckling analysis of Kirchhoff plates with transfer learning ⋮ Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations ⋮ Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks ⋮ fPINNs: Fractional Physics-Informed Neural Networks ⋮ PhyCRNet: physics-informed convolutional-recurrent network for solving spatiotemporal PDEs ⋮ Integrated Lagrangian and Eulerian 3D microstructure-explicit simulations for predicting macroscopic probabilistic SDT thresholds of energetic materials ⋮ A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations ⋮ Overcoming the curse of dimensionality for some Hamilton-Jacobi partial differential equations via neural network architectures ⋮ Efficient coupled deep neural networks for the time-dependent coupled Stokes-Darcy problems ⋮ Learning and meta-learning of stochastic advection–diffusion–reaction systems from sparse measurements ⋮ B-DeepONet: an enhanced Bayesian deeponet for solving noisy parametric PDEs using accelerated replica exchange SGLD ⋮ Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning ⋮ Learning stochastic dynamics with statistics-informed neural network

Uses Software

• TensorFlow
• Adam
• SegNet

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations. II: Adaptivity and generalizations
• A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations. I: Derivation and algorithms
• Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
• Dynamically orthogonal field equations for continuous stochastic dynamical systems
• Machine learning of linear differential equations using Gaussian processes
• Hidden physics models: machine learning of nonlinear partial differential equations
• Bayesian deep convolutional encoder-decoder networks for surrogate modeling and uncertainty quantification
• Bayesian learning for neural networks
• Adaptive multi-element polynomial chaos with discrete measure: algorithms and application to SPDEs
• Inverse problems: A Bayesian perspective
• Numerical Gaussian Processes for Time-Dependent and Nonlinear Partial Differential Equations
• Polynomial Chaos in Stochastic Finite Elements
• Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures
• The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
• Bayesian Graphical Models for Discrete Data

This page was built for publication: Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems