Deep learning of parameterized equations with applications to uncertainty quantification
arXiv:1910.07096
Abstract
We propose a numerical method for discovering unknown parameterized dynamical systems by using observational data of the state variables. Our method is built upon and extends the recent work of discovering unknown dynamical systems, in particular those using deep neural network (DNN). We propose a DNN structure, largely based upon the residual network (ResNet), to not only learn the unknown form of the governing equation but also take into account the random effect embedded in the system, which is generated by the random parameters. Once the DNN model is successfully constructed, it is able to produce system prediction over longer term and for arbitrary parameter values. For uncertainty quantification, it allows us to conduct uncertainty analysis by evaluating solution statistics over the parameter space.
References in corpus (19)
- Adam: A Method for Stochastic Optimization
- Deep Learning in Neural Networks: An Overview
- Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
- DGM: A deep learning algorithm for solving partial differential equations
- Solving high-dimensional partial differential equations using deep learning
- Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data
- Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification
- Chaos as an Intermittently Forced Linear System
- Machine Learning of Linear Differential Equations using Gaussian Processes
- PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network
- Neural Ordinary Differential Equations
- Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification
- Adversarial Uncertainty Quantification in Physics-Informed Neural Networks
- Data Driven Governing Equations Approximation Using Deep 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
- Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems
- Efficient inference of parsimonious phenomenological models of cellular dynamics using S-systems and alternating regression
- Numerical Aspects for Approximating Governing Equations Using Data