1. Extracting Interpretable Physical Parameters from Spatiotemporal Systems Using Unsupervised Learning
    Lu, Peter Y., Kim, Samuel, and Soljačić, Marin
    Phys. Rev. X 2020

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.

Extracting relevant parameters governing the dynamics of spatiotemporal data (with uncontrolled variables), and learning a tunable, transferable predictive model.

Architecture

We implement a physics-informed architecture based on variational autoencoders (VAEs) 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.

Physics-informed VAE architecture.

Experiments

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. In particular, we generate datasets using the 1D Kuramoto–Sivashinsky (KS) equation (with varying viscosity damping parameter γ\gamma acting as an uncontrolled variable) ut=γx4ux2uuxu, \frac{\partial u}{\partial t} = -\gamma \partial_x^4 u - \partial_x^2 u - u\partial_x u, the 1D nonlinear Schrödinger (NS) equation (with varying nonlinearity coefficient κ\kappa) iψt=12x2ψ+κψ2ψ, i \frac{\partial \psi}{\partial t} = -\frac{1}{2} \partial_x^2\psi + \kappa|\psi|^2\psi, and the 2D convection–diffusion (CD) equation (with varying diffusion constant DD and drift velocity v\mathbf{v}) ut=D2uvu. \frac{\partial u}{\partial t} = D\nabla^2u - \mathbf{v} \cdot \nabla u.

PDE prediction examples using the learned predictive models.

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).

Identifying relevant parameters, which have large variance of µz (blue) and small mean of σz2(red).

These extracted parameters correlate well (linearly with R2 > 0.95) with the ground truth physical parameters used to generate the datasets.

Predicted relevant parameters (extracted from the latent parameters of the VAE by a linear fit) vs. the ground truth physical parameters.

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.

Application to nonlinear optical fiber propagation.

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.

Please see our paper for more details.