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 developing innovative methods for solving partial differential equations (PDEs) efficiently and accurately. Recent developments have led to the creation of hybrid frameworks that combine neural networks with traditional numerical methods, allowing for faster adaptation to new PDEs and improved generalization across different domains. Notable advancements include the integration of PDE residuals into pre-training, constraint-aware pre-training, and the use of epistemic PINNs to quantify uncertainty. These innovations have the potential to revolutionize the field of scientific computing and enable the solution of complex PDEs in a wide range of applications. Noteworthy papers in this area include: Physics-Informed Deep B-Spline Networks for Dynamical Systems, which proposes a hybrid framework for solving PDEs with varying parameters and initial conditions. Paving the way for scientific foundation models, which introduces a constraint-aware pre-training method for enhancing generalization and robustness in PDEs. E-PINNs: Epistemic Physics-Informed Neural Networks, which presents a framework for efficiently quantifying uncertainty in PINNs.

Sources

Physics-Informed Deep B-Spline Networks for Dynamical Systems

Physics-Informed Neural Network Surrogate Models for River Stage Prediction

Adaptive Physics-informed Neural Networks: A Survey

Application of Physics-Informed Neural Networks for Solving the Inverse Advection-Diffusion Problem to Localize Pollution Sources

Paving the way for scientific foundation models: enhancing generalization and robustness in PDEs with constraint-aware pre-training

Least Squares with Equality constraints Extreme Learning Machines for the resolution of PDEs

E-PINNs: Epistemic Physics-Informed Neural Networks

Physics-Informed Neural Networks with Unknown Partial Differential Equations: an Application in Multivariate Time Series

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