Advances in Physics-Informed Neural Networks for Solving Partial Differential Equations

The field of physics-informed neural networks (PINNs) is rapidly advancing, with a focus on improving the accuracy and efficiency of solving partial differential equations (PDEs). Recent developments have highlighted the importance of incorporating physical constraints and geometry awareness into neural network architectures. This has led to the development of novel methods, such as the use of liquid residual blocks and measurement-aware cross attention mechanisms, which have shown significant improvements in predictive accuracy and robustness. Notably, the introduction of physics-informed loss functions and geometry-aware spatio-spectral graph neural operators has enabled the solution of complex PDEs with high accuracy and efficiency. Furthermore, the application of PINNs to real-world problems, such as bearing fault classification and satellite attitude dynamics, has demonstrated their potential for practical impact.

Noteworthy papers include: Fast, Convex and Conditioned Network for Multi-Fidelity Vectors and Stiff Univariate Differential Equations, which introduces a simple yet effective activation filtering step to increase matrix rank and expressivity while preserving convexity. Diffeomorphic Neural Operator Learning, which presents an operator learning approach for a class of evolution operators using a composition of a learned lift into the space of diffeomorphisms of the domain and the group action on the field space.

Sources

Fast, Convex and Conditioned Network for Multi-Fidelity Vectors and Stiff Univariate Differential Equations

Diffeomorphic Neural Operator Learning

Onsager Principle-Based Domain Embedding for Thermodynamically Consistent Cahn-Hilliard Model in Arbitrary Domain

Unsupervised operator learning approach for dissipative equations via Onsager principle

Physics-Informed Multimodal Bearing Fault Classification under Variable Operating Conditions using Transfer Learning

Learning Satellite Attitude Dynamics with Physics-Informed Normalising Flow

Prediction error certification for PINNs: Theory, computation, and application to Stokes flow

UGM2N: An Unsupervised and Generalizable Mesh Movement Network via M-Uniform Loss

LNN-PINN: A Unified Physics-Only Training Framework with Liquid Residual Blocks

CVCM Track Circuits Pre-emptive Failure Diagnostics for Predictive Maintenance Using Deep Neural Networks

Open-Set Fault Diagnosis in Multimode Processes via Fine-Grained Deep Feature Representation

Physics-guided Deep Unfolding Network for Enhanced Kronecker Compressive sensing

Physics- and geometry-aware spatio-spectral graph neural operator for time-independent and time-dependent PDEs

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