[论文解读] Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations

Liu Yang, Dongkun Zhang+1 more|arXiv (Cornell University)|Nov 5, 2018

Model Reduction and Neural Networks参考文献 32被引用 59

一句话总结

PI-GANs 将随机微分方程编码到 GAN 中，以在有限的散布数据下解决前向、逆向和混合 SDE 问题，利用 WGAN-GP 提高稳定性。该框架能够在不遇到维度灾难的情况下处理高维随机问题，并支持多组传感器数据。

ABSTRACT

We developed a new class of physics-informed generative adversarial networks (PI-GANs) to solve in a unified manner forward, inverse and mixed stochastic problems based on a limited number of scattered measurements. Unlike standard GANs relying only on data for training, here we encoded into the architecture of GANs the governing physical laws in the form of stochastic differential equations (SDEs) using automatic differentiation. In particular, we applied Wasserstein GANs with gradient penalty (WGAN-GP) for its enhanced stability compared to vanilla GANs. We first tested WGAN-GP in approximating Gaussian processes of different correlation lengths based on data realizations collected from simultaneous reads at sparsely placed sensors. We obtained good approximation of the generated stochastic processes to the target ones even for a mismatch between the input noise dimensionality and the effective dimensionality of the target stochastic processes. We also studied the overfitting issue for both the discriminator and generator, and we found that overfitting occurs also in the generator in addition to the discriminator as previously reported. Subsequently, we considered the solution of elliptic SDEs requiring approximations of three stochastic processes, namely the solution, the forcing, and the diffusion coefficient. We used three generators for the PI-GANs, two of them were feed forward deep neural networks (DNNs) while the other one was the neural network induced by the SDE. Depending on the data, we employed one or multiple feed forward DNNs as the discriminators in PI-GANs. Here, we have demonstrated the accuracy and effectiveness of PI-GANs in solving SDEs for up to 30 dimensions, but in principle, PI-GANs could tackle very high dimensional problems given more sensor data with low-polynomial growth in computational cost.

研究动机与目标

• 为由 SDEs 控制的前向、逆向和混合随机问题提供统一的数据驱动方法的动机。
• 将已知物理规律整合到 GAN 架构中，以从有限测量中学习随机项。
• 开发一种可扩展的方法，能够处理高维随机过程和多组传感器。

提出的方法

• 使用两个独立的前馈网络来建模如 k(x;ω) 和 u(x;ω) 等随机过程。
• 通过自动微分对 SDE 和边界算子进行编码，以为 f(x;ω) 和 b(x;ω) 创建诱导神经网络。
• 应用带梯度惩罚的 WGAN-GP 以实现稳定的对抗训练并近似目标随机分布。
• 采用多数据组和判别器来处理来自不同传感器组的数据，并跨组强制物理一致性。
• 使用 Adagrad/Adam 优化器逐步交替训练生成器和判别器，并采用数据驱动的对抗损失。
• 在不改变底层 PI-GAN 结构的前提下，将框架扩展到前向、逆向和混合问题。

实验结果

研究问题

• RQ1 PI-GANs 是否能够从有限且分散的传感器数据中准确学习随机过程和随机系数？
• RQ2 在 GAN 中对 SDE 物理进行编码相比仅数据的 GAN，如何影响稳定性和准确性？
• RQ3 PI-GANs 能否在一个框架内解决前向、逆向和混合随机问题？
• RQ4 在数据对齐性差或异质时，多个传感器组对学习的影响是什么？

主要发现

• WGAN-GP 能在不同相关长度和传感器数量下实现对随机过程的稳定学习和准确的分布匹配。
• 在合适的传感器数据条件下，PI-GANs 可以逼近高达 30 维的随机过程，而不受维度灾难的影响。
• 判别器和生成器都可能过拟合，包括生成器，这突出了对训练和数据管理的谨慎需求。
• 该方法在均值和随机解的标准差以及扩散系数方面，与基准结果高度一致。
• 使用具有单独判别器的多数据组允许在无需对齐的情况下利用多样的数据源。
• 在目标分布集中于低维流形的情形（例如固定边界情形）下，WGAN-GP 的表现优于普通 GAN。