DeepMoD: Deep learning for Model Discovery in noisy data
This provides a robust method for model discovery in noisy, small datasets for researchers in physics and engineering, though it is incremental as it builds on existing sparse regression and deep learning techniques.
The authors tackled the problem of discovering partial differential equations from noisy spatio-temporal data by introducing DeepMoD, a deep learning algorithm that uses sparse regression on a neural network-constructed function library, achieving robustness with as few as 100 samples and up to 75% noise levels, as validated on physical equations and experimental gel electrophoresis data.
We introduce DeepMoD, a Deep learning based Model Discovery algorithm. DeepMoD discovers the partial differential equation underlying a spatio-temporal data set using sparse regression on a library of possible functions and their derivatives. A neural network approximates the data and constructs the function library, but it also performs the sparse regression. This construction makes it extremely robust to noise, applicable to small data sets, and, contrary to other deep learning methods, does not require a training set. We benchmark our approach on several physical problems such as the Burgers', Korteweg-de Vries and Keller-Segel equations, and find that it requires as few as $\mathcal{O}(10^2)$ samples and works at noise levels up to $75\%$. Motivated by these results, we apply DeepMoD directly on noisy experimental time-series data from a gel electrophoresis experiment and find that it discovers the advection-diffusion equation describing this system.