AI summary · arXiv: 2304.02282v1 · math.NA
About optimal loss function for training physics-informed neural networks under respecting causality
The essentials
- Machine learning has made significant advancements in the last decade, particularly in deep learning.
- Applications created using machine learning can be comparable or superior to humans in many fields of activity.
- Three categories of auxiliary systems are used for solving scientific problems: those trained using previously obtained data, those leveraging previously established physical laws, and those utilizing both approaches.
- Physics-informed neural networks (PINNs) leverage previously established physical laws to solve problems described by differential equations with initial and boundary conditions.
- The PINN approach involves training a neural network to approximate the dependences of physical values on spatial and temporal coordinates for a given problem.
- Hyperparameters and architecture need to be adapted for each new problem.
- Techniques have been developed to improve the accuracy of neural networks trained using the PINN approach, including reducing a problem described by differential equations with initial and boundary conditions to one described only by differential equations and modifying weighted loss functions respecting causality based on generalized functions derived from these modifications.
- Numerical experiments were carried out for various problems demonstrating the accuracy of these proposed methods.
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