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

2021/09/30 by Shamsulhaq Basir, Inanc Senocak · 94 citations
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Algorithm #Applied mathematics #Artificial intelligence #Artificial neural network #Boundary (topology) #Boundary value problem #Computer science #Fidelity #Fluid Dynamics and Turbulent Flows #Function (biology) #Inverse problem #Mathematical analysis #Mathematical optimization #Mathematics #Model Reduction and Neural Networks #Nuclear Engineering Thermal-Hydraulics #Partial differential equation #Residual #cs.LG #cs.NA #math.NA #physics.comp-ph #physics.flu-dyn

paper · pdf · doi:10.1016/j.jcp.2022.111301

published in Journal of Computational Physics 463, 111301 (Elsevier BV)

openalex publication_date 2022/05/17 · arxiv created 2022/05/21 · arxiv updated 2022/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• ImageNet classification with deep convolutional neural networks
• A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems

Cited by

• Domain Decomposition of Large Neural Network Surrogate Models
• Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometries
• Fundamentals of Regression
• Solution of Incompressible Flow Equations with Physics and Equality Constrained Artificial Neural Networks
• Self-adaptive weighting and sampling for physics-informed neural networks
• NASP-T: A Fuzzy Neuro-Symbolic Transformer for Logic-Constrained Aviation Safety Report Classification
• A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learning
• Conditionally adaptive augmented Lagrangian method for physics-informed learning of forward and inverse problems
• Prediction of acoustic field in 1-D uniform duct with varying mean flow and temperature using neural networks
• PC-SRGAN: Physically Consistent Super-Resolution Generative Adversarial Network for General Transient Simulations
• Physics-informed solution reconstruction in elasticity and heat transfer using the explicit constraint force method
• A variational framework for residual-based adaptivity in neural PDE solvers and operator learning. [europepmc]