PDE Solution Operator

Research on PDE solution operators focuses on developing efficient and accurate machine learning methods to approximate the solution of partial differential equations (PDEs), bypassing the need for traditional, computationally expensive discretization techniques. Current efforts center on neural operator architectures, including deep operator networks and novel designs like Hyena operators, often incorporating techniques like bilevel optimization and physics-informed neural networks to improve accuracy and efficiency. These advancements offer significant potential for accelerating the solution of complex PDE-constrained problems across diverse scientific and engineering domains, particularly those involving high-dimensional uncertainty or large-scale simulations.

Papers

April 27, 2024

BiLO: Bilevel Local Operator Learning for PDE inverse problems
Ray Zirui Zhang, Xiaohui Xie, John S. Lowengrub
Soft Constraint Inverse PDE Problem Bilevel Learning PDE Solution Operator

March 19, 2024

Neural Parameter Regression for Explicit Representations of PDE Solution Operators
Konrad Mundinger, Max Zimmer, Sebastian Pokutta
Neural Network Physic Informed Neural Network Operator Learning Deep Regression Solution Operator PDE Solution Operator

June 28, 2023

Hyena Neural Operator for Partial Differential Equations
Saurabh Patil, Zijie Li, Amir Barati Farimani
Neural Operator Partial Differential Equation PDE Solution Operator

May 31, 2023

Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operators
Dingcheng Luo, Thomas O'Leary-Roseberry, Peng Chen, Omar Ghattas
High Efficiency Optimization Purpose High Dimensional Large Scale Neural Operator PDE Solution Operator High Dimensional Uncertainty

January 28, 2023

Deep Operator Learning Lessens the Curse of Dimensionality for PDEs
Ke Chen, Chunmei Wang, Haizhao Yang
Deep Neural Network Operator Learning Data Dimensionality Reversal Curse Discretization Invariant PDE Solution Operator

November 9, 2021

Solving PDE-constrained Control Problems Using Operator Learning
Rakhoon Hwang, Jae Yong Lee, Jin Young Shin, Hyung Ju Hwang
Optimal Control Operator Learning Control Problem Solving Non Rectangular PDE Solution Operator