**Physics-informed Neural Networks (PINN)
**

Table of Contents

# 1. Why Deep Learning Needs Physics?Â¶

Why do data-driven â€˜black-boxâ€™ methods fail?

- May output result that is physically inconsistent
- Easy to learn spurious relationships that look good only on training and test data
- Can lead to poor generalization outside the available data (out-of-sample prediction tasks)

- Interpretability is absent
- Discovering the mechanism of an underlying process is crucial for scientific advancements

Physics-Informed Neural Networks (PINNs)

- Take full advantage of data science methods with the accumulated prior knowledge of scientific theories $\rightarrow$ Improve predictive performance
- Integration of domain knowledge to overcome the issue of imbalanced data & data shortage

## 1.1. Taxonomy of Informed Deep LearningÂ¶

## 1.2. Multilayer Feedforward Networks are Universal ApproximatorsÂ¶

- The Universal Approximation Theorem
- Neural Networks are capable of approximating any Borel measurable function
- Neural Networks (1989)

## 1.3. Neural Networks for Solving Differential EquationsÂ¶

- Neural Algorithm for Solving Differential Equations
- Journal of Computational Physics (1990)
- Neural minimization for finite difference equation

- ANN for ODE and PDE
- IEEE on Neural Networks (1998)