The Neural Network Approach to Inverse Problems in Differential Equations

Abstract

We proposed a framework for solving inverse problems in differential equations based on neural networks and automatic differentiation. Neural networks are used to approximate hidden fields. We analyze the source of errors in the framework and derive an error estimate for a model diffusion equation problem. Besides, we propose a way for sensitivity analysis, utilizing the automatic differentiation mechanism embedded in the framework. It frees people from the tedious and error-prone process of deriving the gradients. Numerical examples exhibit consistency with the convergence analysis and error saturation is noteworthily predicted. We also demonstrate the unique benefits neural networks offer at the same time: universal approximation ability, regularizing the solution, bypassing the curse of dimensionality and leveraging efficient computing frameworks.

arXiv comment32 pages, 9 figures

References in corpus (11)

• DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamics
• MXNet: A Flexible and Efficient Machine Learning Library for Heterogeneous Distributed Systems
• fPINNs: Fractional Physics-Informed Neural Networks
• Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations
• Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems
• Burgers Turbulence
• Understanding and Enhancing the Transferability of Adversarial Examples
• Neural-net-induced Gaussian process regression for function approximation and PDE solution
• Limits of End-to-End Learning
• Geophysical tomography in engineering geology: an overview
• Calibrating Multivariate Lévy Processes with Neural Networks

Cited by in corpus (18)

• Surrogate Modeling for Fluid Flows Based on Physics-Constrained Deep Learning Without Simulation Data
• Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport
• Efficient training of physics-informed neural networks via importance sampling
• Learning Constitutive Relations from Indirect Observations Using Deep Neural Networks
• An artificial neural network framework for reduced order modeling of transient flows
• Solving inverse-PDE problems with physics-aware neural networks
• On the Use of Neural Networks for Full Waveform Inversion
• Short-term traffic prediction using physics-aware neural networks
• ADCME: Learning Spatially-varying Physical Fields using Deep Neural Networks
• Inverse Modeling of Viscoelasticity Materials using Physics Constrained Learning
• Training multi-objective/multi-task collocation physics-informed neural network with student/teachers transfer learnings
• Transfer Learning Enhanced Full Waveform Inversion
• Solving Irregular and Data-enriched Differential Equations using Deep Neural Networks
• Calibrating Multivariate Lévy Processes with Neural Networks
• Geodesy of irregular small bodies via neural density fields: geodesyNets
• Study of the asteroid Bennu using geodesyANNs and Osiris-Rex data
• A Convolutional Neural Network-based Approach to Field Reconstruction
• Trust Region Method for Coupled Systems of PDE Solvers and Deep Neural Networks