从数据中自动发现控制方程的神经网络方法。
Mechanistic PDE Networks for Discovery of Governing Equations
- 用神经网络隐空间表示时空数据的偏微分方程
- 可有效识别复杂场景下的非线性偏微分方程
- 适合需要建模动态过程的研究者使用
我们提出机制型偏微分方程网络（Mechanistic PDE Networks），用于从数据中发现控制方程。该模型将时空数据表示为神经网络隐空间中的时空依赖线性偏微分方程，通过求解并解码这些方程完成特定任务。学习到的偏微分方程表示能自然刻画数据中的时空动态，提升动力学建模能力。然而，高效求解此类方程是重大挑战。为此，我们开发了一个原生支持GPU、并行、稀疏且可微的多网格求解器，专用于线性偏微分方程，作为机制型偏微分方程网络中的模块。借助该求解器，我们提出一种发现架构，可在复杂场景下发现非线性偏微分方程，并对噪声具有鲁棒性。我们在反应-扩散方程和纳维-斯托克斯方程等多个方程上验证了该方法的有效性。
原文摘要 · Abstract (English)
We present Mechanistic PDE Networks -- a model for discovery of governing partial differential equations from data. Mechanistic PDE Networks represent spatiotemporal data as space-time dependent linear partial differential equations in neural network hidden representations. The represented PDEs are then solved and decoded for specific tasks. The learned PDE representations naturally express the spatiotemporal dynamics in data in neural network hidden space, enabling increased power for dynamical modeling. Solving the PDE representations in a compute and memory-efficient way, however, is a significant challenge. We develop a native, GPU-capable, parallel, sparse, and differentiable multigrid solver specialized for linear partial differential equations that acts as a module in Mechanistic PDE Networks. Leveraging the PDE solver, we propose a discovery architecture that can discover nonlinear PDEs in complex settings while also being robust to noise. We validate PDE discovery on a number of PDEs, including reaction-diffusion and Navier-Stokes equations.
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