Enhancing PDE Solving Efficiency with Pre-trained Models and Conservation Laws in PINNs

The recent advancements in the field of Physics-Informed Neural Networks (PINNs) and related neural operator techniques have significantly enhanced the efficiency and accuracy of solving partial differential equations (PDEs). A notable trend is the integration of pre-trained models, such as DeepONets, into PINNs to reduce training time and computational cost while maintaining high accuracy. This approach, often referred to as fine-tuning, has shown promising results in various types of PDEs, including those with complex boundary conditions and initial states. Additionally, there is a growing focus on enforcing conservation laws within PINNs to ensure physical consistency, which is crucial for modeling real-world systems. Multimodal policies and physics-informed representations are also emerging as key strategies for handling sparse and uncertain observations in control problems involving PDE systems. Furthermore, the application of neural networks to long-time integration of nonlinear wave equations and the approximation of Green's functions through multiscale neural networks are areas that are gaining traction. These developments collectively point towards a more robust and efficient framework for solving PDEs, leveraging the strengths of both traditional numerical methods and modern machine learning techniques.

Noteworthy papers include one that introduces a novel projection method to enforce conservation laws in PINNs, significantly improving adherence to physical laws, and another that proposes a hybrid approach combining PINNs with cylindrical approximations for functional differential equations, offering convergence guarantees and scalability improvements.

Sources

Fine-Tuning DeepONets to Enhance Physics-informed Neural Networks for solving Partial Differential Equations

Physics Informed Neural Networks for heat conduction with phase change

Design and Implementation of Hedge Algebra Controller using Recursive Semantic Values for Cart-pole System

A Least-Squares-Based Neural Network (LS-Net) for Solving Linear Parametric PDEs

Multimodal Policies with Physics-informed Representations

Long-time Integration of Nonlinear Wave Equations with Neural Operators

Residues in Partial Fraction Decomposition Applied to Pole Sensitivity Analysis and Root Locus Construction

Iterative Cut-Based PWA Approximation of Multi-Dimensional Nonlinear Systems

Deep Uzawa for PDE constrained optimisation

Guaranteeing Conservation Laws with Projection in Physics-Informed Neural Networks

Deep learning for model correction of dynamical systems with data scarcity

Physics-informed Neural Networks for Functional Differential Equations: Cylindrical Approximation and Its Convergence Guarantees

Multiscale Neural Networks for Approximating Green's Functions

Differential Informed Auto-Encoder

Using Parametric PINNs for Predicting Internal and External Turbulent Flows

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