STENCIL-NET: Data-driven solution-adaptive discretization of partial differential equations

Abstract

Numerical methods for approximately solving partial differential equations (PDE) are at the core of scientific computing. Often, this requires high-resolution or adaptive discretization grids to capture relevant spatio-temporal features in the PDE solution, e.g., in applications like turbulence, combustion, and shock propagation. Numerical approximation also requires knowing the PDE in order to construct problem-specific discretizations. Systematically deriving such solution-adaptive discrete operators, however, is a current challenge. Here we present STENCIL-NET, an artificial neural network architecture for data-driven learning of problem- and resolution-specific local discretizations of nonlinear PDEs. STENCIL-NET achieves numerically stable discretization of the operators in an unknown nonlinear PDE by spatially and temporally adaptive parametric pooling on regular Cartesian grids, and by incorporating knowledge about discrete time integration. Knowing the actual PDE is not necessary, as solution data is sufficient to train the network to learn the discrete operators. A once-trained STENCIL-NET model can be used to predict solutions of the PDE on larger spatial domains and for longer times than it was trained for, hence addressing the problem of PDE-constrained extrapolation from data. To support this claim, we present numerical experiments on long-term forecasting of chaotic PDE solutions on coarse spatio-temporal grids. We also quantify the speed-up achieved by substituting base-line numerical methods with equation-free STENCIL-NET predictions on coarser grids with little compromise on accuracy.

References in corpus (10)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations
• Using Machine Learning to Replicate Chaotic Attractors and Calculate Lyapunov Exponents from Data
• PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network
• Learning data driven discretizations for partial differential equations
• Accelerating Eulerian Fluid Simulation With Convolutional Networks
• Deep learning of dynamics and signal-noise decomposition with time-stepping constraints
• Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems
• Learned discretizations for passive scalar advection in a 2-D turbulent flow
• Stability selection enables robust learning of partial differential equations from limited noisy data