# Extracting Interpretable Physical Parameters from Spatiotemporal Systems using Unsupervised Learning

## TL;DR

This paper presents an unsupervised, physics-informed deep learning approach using variational autoencoders to extract interpretable physical parameters from noisy spatiotemporal data, enabling better understanding of complex systems governed by PDEs.

## Contribution

The authors develop a novel variational autoencoder architecture tailored for PDE-governed systems, capable of accurately identifying and extracting physical parameters from noisy, real-world data.

## Key findings

- Accurately identifies physical parameters with over 95% correlation to ground truth.
- Effective on both simulated PDE data and realistic fiber propagation data.
- Robust to approximately 10% noise in the input data.

## Abstract

Experimental data is often affected by uncontrolled variables that make analysis and interpretation difficult. For spatiotemporal systems, this problem is further exacerbated by their intricate dynamics. Modern machine learning methods are particularly well-suited for analyzing and modeling complex datasets, but to be effective in science, the result needs to be interpretable. We demonstrate an unsupervised learning technique for extracting interpretable physical parameters from noisy spatiotemporal data and for building a transferable model of the system. In particular, we implement a physics-informed architecture based on variational autoencoders that is designed for analyzing systems governed by partial differential equations (PDEs). The architecture is trained end-to-end and extracts latent parameters that parameterize the dynamics of a learned predictive model for the system. To test our method, we train our model on simulated data from a variety of PDEs with varying dynamical parameters that act as uncontrolled variables. Numerical experiments show that our method can accurately identify relevant parameters and extract them from raw and even noisy spatiotemporal data (tested with roughly 10% added noise). These extracted parameters correlate well (linearly with $R^2 > 0.95$) with the ground truth physical parameters used to generate the datasets. We then apply this method to nonlinear fiber propagation data, generated by an ab-initio simulation, to demonstrate its capabilities on a more realistic dataset. Our method for discovering interpretable latent parameters in spatiotemporal systems will allow us to better analyze and understand real-world phenomena and datasets, which often have unknown and uncontrolled variables that alter the system dynamics and cause varying behaviors that are difficult to disentangle.