论文标题
蒙特卡洛·皮恩（Monte Carlo Pinns）：涉及高维部分偏微分方程的前进和反问题的深度学习方法
Monte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equations
论文作者
论文摘要
我们引入了基于抽样的机器学习方法，蒙特卡洛物理学知情的神经网络（MC-PINNS），用于求解前向和反向分数偏微分方程（FPDES）。作为物理知识的神经网络（PINN）的概括，我们的方法还依赖于深度神经网络替代物，除了计算DNN输出的分数衍生物的随机近似策略外。 MC-Pinns中的一个关键成分是构建对损耗函数中物理软约束的无偏估计。与在\ cite {pang2019fpinns}中提出的FPINS相比，我们的直接采样方法可以产生较低的总体计算成本，因此为解决高维分数PDE提供了机会。我们通过几个示例来验证MC-PINNS方法的性能，包括高维积分拉普拉斯方程，时空分数PDE的参数识别以及带随机输入的分数扩散方程。结果表明，MC-Pinns是灵活的，并且有望解决高维FPDES。
We introduce a sampling based machine learning approach, Monte Carlo physics informed neural networks (MC-PINNs), for solving forward and inverse fractional partial differential equations (FPDEs). As a generalization of physics informed neural networks (PINNs), our method relies on deep neural network surrogates in addition to a stochastic approximation strategy for computing the fractional derivatives of the DNN outputs. A key ingredient in our MC-PINNs is to construct an unbiased estimation of the physical soft constraints in the loss function. Our directly sampling approach can yield less overall computational cost compared to fPINNs proposed in \cite{pang2019fpinns} and thus provide an opportunity for solving high dimensional fractional PDEs. We validate the performance of MC-PINNs method via several examples that include high dimensional integral fractional Laplacian equations, parametric identification of time-space fractional PDEs, and fractional diffusion equation with random inputs. The results show that MC-PINNs is flexible and promising to tackle high-dimensional FPDEs.
