FiniteNet: A Fully Convolutional LSTM Network Architecture for Time-Dependent Partial Differential Equations

In this work, we present a machine learning approach for reducing the error when numerically solving time-dependent partial differential equations (PDE). We use a fully convolutional LSTM network to exploit the spatiotemporal dynamics of PDEs. The neural network serves to enhance finite-difference and finite-volume methods (FDM/FVM) that are commonly used to solve PDEs, allowing us to maintain guarantees on the order of convergence of our method. We train the network on simulation data, and show that our network can reduce error by a factor of 2 to 3 compared to the baseline algorithms. We demonstrate our method on three PDEs that each feature qualitatively different dynamics. We look at the linear advection equation, which propagates its initial conditions at a constant speed, the inviscid Burgers' equation, which develops shockwaves, and the Kuramoto-Sivashinsky (KS) equation, which is chaotic.

Enhancement of shock-capturing methods via machine learning

In recent years, machine learning has been used to create data-driven solutions to problems for which an algorithmic solution is intractable, as well as fine-tuning existing algorithms. This research applies machine learning to the development of an improved finite-volume method for simulating PDEs with discontinuous solutions. Shock capturing methods make use of nonlinear switching functions that are not guaranteed to be optimal. Because data can be used to learn nonlinear relationships, we train a neural network to improve the results of a fifth-order WENO method. We post-process the outputs of the neural network to guarantee that the method is consistent. The training data consists of the exact mapping between cell averages and interpolated values for a set of integrable functions that represent waveforms we would expect to see while simulating a PDE. We demonstrate our method on linear advection of a discontinuous function, the inviscid Burgers' equation, and the 1-D Euler equations. For the latter, we examine the Shu-Osher model problem for turbulence-shockwave interactions. We find that our method outperforms WENO in simulations where the numerical solution becomes overly diffused due to numerical viscosity.