Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion

Abstract

Physics-informed neural networks (PINNs) have been proposed to learn the solution of partial differential equations (PDE). In PINNs, the residual form of the PDE of interest and its boundary conditions are lumped into a composite objective function as soft penalties. Here, we show that this specific way of formulating the objective function is the source of severe limitations in the PINN approach when applied to different kinds of PDEs. To address these limitations, we propose a versatile framework based on a constrained optimization problem formulation, where we use the augmented Lagrangian method (ALM) to constrain the solution of a PDE with its boundary conditions and any high-fidelity data that may be available. Our approach is adept at forward and inverse problems with multi-fidelity data fusion. We demonstrate the efficacy and versatility of our physics- and equality-constrained deep-learning framework by applying it to several forward and inverse problems involving multi-dimensional PDEs. Our framework achieves orders of magnitude improvements in accuracy levels in comparison with state-of-the-art physics-informed neural networks.

References in corpus (20)

• Adam: A Method for Stochastic Optimization
• Deep Learning in Neural Networks: An Overview
• Neural Machine Translation by Jointly Learning to Align and Translate
• Sequence to Sequence Learning with Neural Networks
• TensorFlow: A system for large-scale machine learning
• ADADELTA: An Adaptive Learning Rate Method
• Machine Learning for Fluid Mechanics
• Artificial Neural Networks for Solving Ordinary and Partial Differential Equations
• 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
• A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems
• Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks
• Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations
• Deep Learning of Vortex Induced Vibrations
• Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations
• Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism
• SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs
• Hierarchical Deep Learning of Multiscale Differential Equation Time-Steppers
• Training neural networks under physical constraints using a stochastic augmented Lagrangian approach

Cited by in corpus (14)

• Residual-based attention in physics-informed neural networks
• Estimating density, velocity, and pressure fields in supersonic flow using physics-informed BOS
• Optimizing a DIscrete Loss (ODIL) to solve forward and inverse problems for partial differential equations using machine learning tools
• Investigating and Mitigating Failure Modes in Physics-informed Neural Networks (PINNs)
• Self-adaptive weights based on balanced residual decay rate for physics-informed neural networks and deep operator networks
• Surrogate modelling and uncertainty quantification based on multi-fidelity deep neural network
• I-FENN for thermoelasticity based on physics-informed temporal convolutional network (PI-TCN)
• Learning in PINNs: Phase transition, diffusion equilibrium, and generalization
• Neural network based approach for solving problems in plane wave duct acoustics
• Multi-Fidelity Machine Learning Applied to Steady Fluid Flows
• Neural topology optimization: the good, the bad, and the ugly
• Solving 2-D Helmholtz equation in the rectangular, circular, and elliptical domains using neural networks
• Constrained or Unconstrained? Neural-Network-Based Equation Discovery from Data
• Prediction of acoustic field in 1-D uniform duct with varying mean flow and temperature using neural networks