The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approach
Not on the map. Why?
Abstract (arXiv)
We study the combination of the alternating direction method of multipliers (ADMM) with physics-informed neural networks (PINNs) for a general class of nonsmooth partial differential equation (PDE)-constrained optimization problems, where additional regularization can be employed for constraints on the control or design variables. The resulting ADMM-PINNs algorithmic framework substantially enlarges the applicable range of PINNs to nonsmooth cases of PDE-constrained optimization problems. The application of the ADMM makes it possible to untie the PDE constraints and the nonsmooth regularization terms for iterations. Accordingly, at each iteration, one of the resulting subproblems is a smooth PDE-constrained optimization which can be efficiently solved by PINNs, and the other is a simple nonsmooth optimization problem which usually has a closed-form solution or can be efficiently solved by various standard optimization algorithms or pre-trained neural networks. The ADMM-PINNs algorithmic framework does not require to solve PDEs repeatedly, and it is mesh-free, easy to implement, and scalable to different PDE settings. We validate the efficiency of the ADMM-PINNs algorithmic framework by different prototype applications, including inverse potential problems, source identification in elliptic equations, control constrained optimal control of the Burgers equation, and sparse optimal control of parabolic equations.
Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 39 of its 95 references that are in the map. First posted on arXiv 16 Feb 2023.
Builds on
39 of 95 references.
References · 39
- 2016Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoisingcited by 1,352
- 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
- 2019DeepXDE: A deep learning library for solving differential equationscited by 617
- 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
- 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
- 2017Solving high-dimensional partial differential equations using deep learningcited by 549
- 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
- 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
- 2017The Expressive Power of Neural Networks: A View from the Widthcited by 318
- 2016On the Expressive Power of Deep Neural Networkscited by 302
- 2021Physics-informed neural networks with hard constraints for inverse designcited by 277
- 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 261
27 more
- 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 220
- 2022Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisonscited by 169
- 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 144
- 2019Variational Physics-Informed Neural Networks For Solving Partial Differential Equationscited by 144
- 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 128
- 2021CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Methodcited by 93
- 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 88
- 2020Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equationscited by 78
- 2017Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equationscited by 72
- 2019Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equationscited by 66
- 2017Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equationscited by 62
- 2021Optimal control of PDEs using physics-informed neural networkscited by 51
- 2021Solving PDE-constrained Control Problems Using Operator Learningcited by 41
- 2021Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusioncited by 38
- 2022NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operatorscited by 33
- 2020Solving inverse-PDE problems with physics-aware neural networkscited by 27
- 2020Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networkscited by 26
- 2021Hybrid FEM-NN models: Combining artificial neural networks with the finite element methodcited by 24
- 2021Fast PDE-constrained optimization via self-supervised operator learningcited by 24
- 2022Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computingcited by 24
- 2018Neural-net-induced Gaussian process regression for function approximation and PDE solutioncited by 22
- 2022Physics-informed neural networks for PDE-constrained optimization and controlcited by 19
- 2022Bi-level Physics-Informed Neural Networks for PDE Constrained Optimization using Broyden's Hypergradientscited by 17
- 2022Neural Q-learning for solving PDEscited by 8
- 2022A two-stage numerical approach for the sparse initial source identification of a diffusion-advection equationcited by 3
- 2021Control and numerical approximation of fractional diffusion equationscited by 1
- 2022A numerical approach to the optimal control of thermally convective flowscited by 1
Built on by
Not cited yet.