PDE-LEARN: Using deep learning to discover partial differential equations from noisy, limited data.

In this paper, we introduce PDE-LEARN, a novel deep learning algorithm that can identify governing partial differential equations (PDEs) directly from noisy, limited measurements of a physical system of interest. PDE-LEARN uses a Rational Neural Network, U, to approximate the system response function and a sparse, trainable vector, ξ, to characterize the hidden PDE that the system response function satisfies. Our approach couples the training of U and ξ using a loss function that (1) makes U approximate the system response function, (2) encapsulates the fact that U satisfies a hidden PDE that ξ characterizes, and (3) promotes sparsity in ξ using ideas from iteratively reweighted least-squares. Further, PDE-LEARN can simultaneously learn from several data sets, allowing it to incorporate results from multiple experiments. This approach yields a robust algorithm to discover PDEs directly from realistic scientific data. We demonstrate the efficacy of PDE-LEARN by identifying several PDEs from noisy and limited measurements.

PDE-READ: Human-readable partial differential equation discovery using deep learning.

PDE discovery shows promise for uncovering predictive models of complex physical systems but has difficulty when measurements are noisy and limited. We introduce a new approach for PDE discovery that uses two Rational Neural Networks and a principled sparse regression algorithm to identify the hidden dynamics that govern a system's response. The first network learns the system response function, while the second learns a hidden PDE describing the system's evolution. We then use a parameter-free sparse regression algorithm to extract a human-readable form of the hidden PDE from the second network. We implement our approach in an open-source library called PDE-READ. Our approach successfully identifies the governing PDE in six benchmark examples. We demonstrate that our approach is robust to both sparsity and noise and it, therefore, holds promise for application to real-world observational data.