**Physics-Guided Machine Learning**

Work in progress

In what follows, we distinguish two cases wether the PDE describing the underlying dynamic is known or not.

## 1. Models

**PINNs** [Raissi et al., 2018]
PDE-known. This paper first introduced the concept of *Physics-Informed Neural Networks* (PINNs). Instead of resorting to numerical PDE solvers, the originality of this work is to train a neural network to predict the solution of a known PDE at any spatio-temporal point. More formally, it relies on a neural network \(f_\theta\), with parameters \(\theta\), whose goal is to map \([x,t]\) to the solution \(u([x,t])\) of the PDE at \([x,t]\). The parameters \(\theta\) are trained so that three conditions are met:

- The PDE is satisfied
- The boundary conditions (BC) are satisfied
- The initial conditions (IC) are satisfied

In practice, \(\ell_2^2\) residuals (i.e., MSEs) are computed on some pre-defined spatio-temporal points to quantify, to what extents, each of the condition is satisfied. As such, \(\theta\) is found so that the sum of \(\mathcal{L}_{\rm PDE}\), \(\mathcal{L}_{\rm BC}\) and \(\mathcal{L}_{\rm IC}\) is minimized

**APHYNITY** [Yin et al., 2021]
PDE-partly-known.

**FNO** [Li et al., 2021]
PDE-unknown. *Fourier Neural Operators* (FNO)

## 2. Meta-Learning

**CoDA** [Kirchmeyer et al., 2022]
PDE-unknown. The *COntext-informed Dynamics Adaptation* (CoDA) method …

## References

**[Kirchmeyer et al., 2022]**M. Kirchmeyer, Y. Yin, J. Donà, N. Baskiotis, A. Rakotomamonjy and P. Gallinari. "Generalizing to new physical systems via context-informed dynamics model". Proceedings of the International Conference on Machine Learning (ICML) (2022)**[Kirchmeyer et al., 2022]**M. Kirchmeyer, Y. Yin, J. Donà, N. Baskiotis, A. Rakotomamonjy and P. Gallinari. "Generalizing to new physical systems via context-informed dynamics model". Proceedings of the International Conference on Machine Learning (ICML) (2022)**[Raissi et al., 2018]**M. Raissi, P. Perdikaris and G.E. Karniadakis. "Physics-informed neural networks - a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations". Journal of Computational Physics (2018)**[Yin et al., 2021]**Y. Yin, V. Le Guen, J. Dona, E. de Bézenac, I. Ayed, N. Thome and P. Gallinari. "Augmenting physical models with deep networks for complex dynamics forecasting". Journal of Statistical Mechanics - Theory and Experiment (2021)