Learning continuous-time PDEs from sparse data with graph neural networks
arXiv:2006.08956
Abstract
The behavior of many dynamical systems follow complex, yet still unknown partial differential equations (PDEs). While several machine learning methods have been proposed to learn PDEs directly from data, previous methods are limited to discrete-time approximations or make the limiting assumption of the observations arriving at regular grids. We propose a general continuous-time differential model for dynamical systems whose governing equations are parameterized by message passing graph neural networks. The model admits arbitrary space and time discretizations, which removes constraints on the locations of observation points and time intervals between the observations. The model is trained with continuous-time adjoint method enabling efficient neural PDE inference. We demonstrate the model's ability to work with unstructured grids, arbitrary time steps, and noisy observations. We compare our method with existing approaches on several well-known physical systems that involve first and higher-order PDEs with state-of-the-art predictive performance.
References in corpus (12)
- Neural Message Passing for Quantum Chemistry
- DGM: A deep learning algorithm for solving partial differential equations
- Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations
- Interaction Networks for Learning about Objects, Relations and Physics
- Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks
- Graph networks as learnable physics engines for inference and control
- PDE-Net: Learning PDEs from Data
- Data driven approximation of parametrized PDEs by Reduced Basis and Neural Networks
- Graph Neural Ordinary Differential Equations
- Differentiable Physics-informed Graph Networks
- Neural network augmented inverse problems for PDEs
- DPM: A deep learning PDE augmentation method (with application to large-eddy simulation)