The field of partial differential equations (PDEs) is experiencing significant advancements with the integration of deep learning techniques. Recent developments have introduced innovative methods for solving high-dimensional PDEs, which were previously challenging due to the curse of dimensionality. Quantum-inspired approaches, such as the use of quantized tensor trains, have shown promising results in achieving efficient and accurate solutions. Additionally, the application of neural networks and deep learning architectures has enabled the solution of complex PDEs with improved accuracy and reduced computational cost. Noteworthy papers in this area include the introduction of the Spectral-inspired Neural Operator, which learns PDE operators from limited trajectories, and the development of KITINet, a novel architecture that reinterprets feature propagation through the lens of non-equilibrium particle dynamics and PDE simulation. These advancements have the potential to revolutionize the field of PDEs and enable the solution of complex problems in various disciplines.
Advancements in Solving Partial Differential Equations with Deep Learning
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A brief review of the Deep BSDE method for solving high-dimensional partial differential equations
Fast and Flexible Quantum-Inspired Differential Equation Solvers with Data Integration
Efficient Training of Neural SDEs Using Stochastic Optimal Control
Finite element spaces of double forms
Data-driven Closure Strategies for Parametrized Reduced Order Models via Deep Operator Networks
Multiphysics Bench: Benchmarking and Investigating Scientific Machine Learning for Multiphysics PDEs
Application of troubled-cells to finite volume methods -- an optimality study using a novel monotonicity parameter
A sparse $hp$-finite element method for piecewise-smooth differential equations with periodic boundary conditions
KITINet: Kinetics Theory Inspired Network Architectures with PDE Simulation Approaches
Spectral-inspired Neural Operator for Data-efficient PDE Simulation in Physics-agnostic Regimes
A Physics-Informed Learning Framework to Solve the Infinite-Horizon Optimal Control Problem
Full Domain Analysis in Fluid Dynamics
Multiprecision computing for multistage fractional physics-informed neural networks
FNOPE: Simulation-based inference on function spaces with Fourier Neural Operators
A comprehensive analysis of PINNs: Variants, Applications, and Challenges
Deep asymptotic expansion method for solving singularly perturbed time-dependent reaction-advection-diffusion equations
Neural Interpretable PDEs: Harmonizing Fourier Insights with Attention for Scalable and Interpretable Physics Discovery
DeepRTE: Pre-trained Attention-based Neural Network for Radiative Tranfer
(U)NFV: Supervised and Unsupervised Neural Finite Volume Methods for Solving Hyperbolic PDEs