论文标题
使用神经网络解决方案束求解微分方程
Solving Differential Equations Using Neural Network Solution Bundles
论文作者
论文摘要
动态系统的时间演变经常由普通微分方程（ODE）描述，在给定初始条件下必须求解。大多数标准方法都会集成产生单个解决方案的ODE，该解决方案是在离散时间计算的。当需要许多具有不同初始条件的不同解决方案对ode需要不同，计算成本可能会变得很大。我们建议将神经网络用作解决方案捆绑包，这是各种初始状态和系统参数的ode解决方案的集合。神经网络解决方案捆绑包经过无监督的损失，该损失不需要任何知识的解决方案，并且在初始条件和系统参数中可以差异化。解决方案束显示系统状态的快速，可行的评估，促进了贝叶斯推断对实际动力学系统中参数估计的使用。
The time evolution of dynamical systems is frequently described by ordinary differential equations (ODEs), which must be solved for given initial conditions. Most standard approaches numerically integrate ODEs producing a single solution whose values are computed at discrete times. When many varied solutions with different initial conditions to the ODE are required, the computational cost can become significant. We propose that a neural network be used as a solution bundle, a collection of solutions to an ODE for various initial states and system parameters. The neural network solution bundle is trained with an unsupervised loss that does not require any prior knowledge of the sought solutions, and the resulting object is differentiable in initial conditions and system parameters. The solution bundle exhibits fast, parallelizable evaluation of the system state, facilitating the use of Bayesian inference for parameter estimation in real dynamical systems.
