Sparsely constrained neural networks for model discovery of PDEs
This work addresses model discovery for PDEs, enabling applications in physics and engineering with noisy or sparsely sampled data, though it is incremental as it builds on existing sparse regression and neural network methods.
The authors tackled the problem of discovering partial differential equations (PDEs) from spatio-temporal data by introducing a modular framework that dynamically determines sparsity patterns in neural network-based surrogates, improving accuracy and convergence on benchmark examples.
Sparse regression on a library of candidate features has developed as the prime method to discover the partial differential equation underlying a spatio-temporal data-set. These features consist of higher order derivatives, limiting model discovery to densely sampled data-sets with low noise. Neural network-based approaches circumvent this limit by constructing a surrogate model of the data, but have to date ignored advances in sparse regression algorithms. In this paper we present a modular framework that dynamically determines the sparsity pattern of a deep-learning based surrogate using any sparse regression technique. Using our new approach, we introduce a new constraint on the neural network and show how a different network architecture and sparsity estimator improve model discovery accuracy and convergence on several benchmark examples. Our framework is available at \url{https://github.com/PhIMaL/DeePyMoD}