Solution Operator

Solution operators, which map input functions (e.g., initial conditions) to solutions of partial differential equations (PDEs), are a central focus in scientific machine learning. Current research emphasizes developing and improving neural operator architectures, such as Fourier Neural Operators (FNOs), DeepONets, and variations incorporating physics-informed learning or uncertainty quantification, to efficiently and accurately learn these operators from data. This work is significant because it offers faster and more flexible alternatives to traditional numerical PDE solvers, with applications ranging from fluid dynamics and materials science to weather forecasting and control systems. The development of robust and generalizable solution operators promises to accelerate scientific discovery and improve the efficiency of engineering design.

Papers

October 18, 2024

Simultaneously Solving FBSDEs with Neural Operators of Logarithmic Depth, Constant Width, and Sub-Linear Rank
Takashi Furuya, Anastasis Kratsios
Neural Operator Large Depth Elliptic Partial Differential Equation Solution Operator Discrete Logarithm Ranking Consistency Dependent FBSDEs Forward Backward Stochastic Differential Equation

May 11, 2024

Learning Flame Evolution Operator under Hybrid Darrieus Landau and Diffusive Thermal Instability
Rixin Yu, Erdzan Hodzic, Karl-Johan Nogenmyr
Fourier Neural Operator Solution Operator Landau Equation Thermal Fluctuation

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

March 15, 2024

Using Uncertainty Quantification to Characterize and Improve Out-of-Domain Learning for PDEs
S. Chandra Mouli, Danielle C. Maddix, Shima Alizadeh, Gaurav Gupta, Andrew Stuart, Michael W. Mahoney, Yuyang Wang
Uncertainty Quantification Domain Generalization Neural Operator Scientific Machine Learning Solution Operator

February 23, 2024

Learning solution operators of PDEs defined on varying domains via MIONet
Shanshan Xiao, Pengzhan Jin, Yifa Tang
Domain Name Solution Operator Poisson Equation

February 14, 2024

February 11, 2024

DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domains
Minglang Yin, Nicolas Charon, Ryan Brody, Lu Lu, Natalia Trayanova, Mauro Maggioni
Partial Differential Equation Operator Learning Domain Name Solution Operator Time Dependent Partial Differential Equation Multiscale Elliptic

January 27, 2024

Unsupervised Solution Operator Learning for Mean-Field Games via Sampling-Invariant Parametrizations
Han Huang, Rongjie Lai
Mean Field Operator Learning Mean Field Game Solution Operator

January 18, 2024

Approximation of Solution Operators for High-dimensional PDEs
Nathan Gaby, Xiaojing Ye
Average Approximation Reduced Order High Dimensional Partial Differential Equation Solution Operator PDE Control

December 18, 2023

Inducing Point Operator Transformer: A Flexible and Scalable Architecture for Solving PDEs
Seungjun Lee, Taeil Oh
Point Transformer Data Discretization Numerical Discretization Solution Operator Scalable Architecture

October 8, 2023

Waveformer for modelling dynamical systems
N Navaneeth, Souvik Chakraborty
Dynamical System Neural Operator Navier Stokes Solution Operator

August 4, 2023

Learning the solution operator of two-dimensional incompressible Navier-Stokes equations using physics-aware convolutional neural networks
Viktor Grimm, Alexander Heinlein, Axel Klawonn
Solution Operator Finite Difference Discretization Method

June 21, 2023

Residual-Based Error Corrector Operator to Enhance Accuracy and Reliability of Neural Operator Surrogates of Nonlinear Variational Boundary-Value Problems
Prashant K. Jha
Neural Operator Nonlinear Model Variational Method Solution Operator Enhancing Accuracy Operator Surrogate

April 9, 2023

Variational operator learning: A unified paradigm marrying training neural operators and solving partial differential equations
Tengfei Xu, Dachuan Liu, Peng Hao, Bo Wang
Training Data Unified Framework Neural Operator Partial Differential Equation Operator Learning Solution Operator

February 23, 2023

Learning stiff chemical kinetics using extended deep neural operators
Somdatta Goswami, Ameya D. Jagtap, Hessam Babaee, Bryan T. Susi, George Em Karniadakis
Neural Operator Solution Operator Deep Neural Operator Chemical Kinetics Operator Surrogate

January 28, 2023

MetaNO: How to Transfer Your Knowledge on Learning Hidden Physics
Lu Zhang, Huaiqian You, Tian Gao, Mo Yu, Chung-Hao Lee, Yue Yu
Meta Learning Knowledge Based Neural Operator Solution Operator Hidden Physic

January 19, 2023

Forecasting subcritical cylinder wakes with Fourier Neural Operators
Peter I Renn, Cong Wang, Sahin Lale, Zongyi Li, Anima Anandkumar, Morteza Gharib
Predictive Control Neural Operator Fourier Neural Operator Velocity Field Solution Operator Turbulence Model Wake Model

January 17, 2023

Improved generalization with deep neural operators for engineering systems: Path towards digital twin
Kazuma Kobayashi, James Daniell, Syed Bahauddin Alam
Digital Twin Neural Operator Root to Leaf Path Deep Operator Network Improved Generalization Solution Operator Deep Neural Operator Engineering System Nonlinear Operator Mapping Wave Speed

December 9, 2022

A PINN Approach to Symbolic Differential Operator Discovery with Sparse Data
Lena Podina, Brydon Eastman, Mohammad Kohandel
Physic Informed Neural Network Scientific Discovery Sparse Data PINN Method Solution Operator Differential Operator

Solution Operator

Papers

Simultaneously Solving FBSDEs with Neural Operators of Logarithmic Depth, Constant Width, and Sub-Linear Rank

Learning Flame Evolution Operator under Hybrid Darrieus Landau and Diffusive Thermal Instability

Neural Parameter Regression for Explicit Representations of PDE Solution Operators

Using Uncertainty Quantification to Characterize and Improve Out-of-Domain Learning for PDEs

Learning solution operators of PDEs defined on varying domains via MIONet

Sobolev Training for Operator Learning

Neural Operators Meet Energy-based Theory: Operator Learning for Hamiltonian and Dissipative PDEs

DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domains

Unsupervised Solution Operator Learning for Mean-Field Games via Sampling-Invariant Parametrizations

Approximation of Solution Operators for High-dimensional PDEs

Inducing Point Operator Transformer: A Flexible and Scalable Architecture for Solving PDEs

Waveformer for modelling dynamical systems

Learning the solution operator of two-dimensional incompressible Navier-Stokes equations using physics-aware convolutional neural networks

Residual-Based Error Corrector Operator to Enhance Accuracy and Reliability of Neural Operator Surrogates of Nonlinear Variational Boundary-Value Problems

Variational operator learning: A unified paradigm marrying training neural operators and solving partial differential equations

Learning stiff chemical kinetics using extended deep neural operators

MetaNO: How to Transfer Your Knowledge on Learning Hidden Physics

Forecasting subcritical cylinder wakes with Fourier Neural Operators

Improved generalization with deep neural operators for engineering systems: Path towards digital twin

A PINN Approach to Symbolic Differential Operator Discovery with Sparse Data