论文标题
diffusionnet：使用深度学习加速依赖时间相关的偏微分方程的解决方案
DiffusionNet: Accelerating the solution of Time-Dependent partial differential equations using deep learning
论文作者
论文摘要
我们介绍了深度学习框架，以解决和加速一个和两个空间维度的时间依赖性偏微分方程解决方案。我们通过在Dirichlet边界条件下解决2D瞬态热传导问题来证明扩散网络求解器。对使用交替方向隐式方法计算的解决方案数据训练该模型。我们显示了该模型从七个变量的任何组合预测解决方案的能力：解决方案的启动时间步骤，初始条件，四个边界条件以及时间步长大小，扩散率常数和网格步长的组合变量。为了提高速度，我们利用模型能力来一次预测时间依赖性PDE的解决方案在多个时间步骤之后，通过将溶液分为可行的可行块来提高解决方案的速度。我们尝试构建一个灵活的体系结构，能够以最小的变化来求解广泛的部分微分方程。我们通过将模型应用于用于求解瞬态热传导以求解无粘性汉堡方程和稳态热传导的相同网络体系结构的模型来证明我们的模型灵活性，然后将我们的模型性能与相关研究进行比较。我们表明，我们的模型减少了解决问题的解决方案的误差。
We present our deep learning framework to solve and accelerate the Time-Dependent partial differential equation's solution of one and two spatial dimensions. We demonstrate DiffusionNet solver by solving the 2D transient heat conduction problem with Dirichlet boundary conditions. The model is trained on solution data calculated using the Alternating direction implicit method. We show the model's ability to predict the solution from any combination of seven variables: the starting time step of the solution, initial condition, four boundary conditions, and a combined variable of the time step size, diffusivity constant, and grid step size. To improve speed, we exploit our model capability to predict the solution of the Time-dependent PDE after multiple time steps at once to improve the speed of solution by dividing the solution into parallelizable chunks. We try to build a flexible architecture capable of solving a wide range of partial differential equations with minimal changes. We demonstrate our model flexibility by applying our model with the same network architecture used to solve the transient heat conduction to solve the Inviscid Burgers equation and Steady-state heat conduction, then compare our model performance against related studies. We show that our model reduces the error of the solution for the investigated problems.
