论文标题
爱因斯坦的场方程作为连续时间复发的神经网络
Einstein's Field Equations as Continuous-Time Recurrent Neural Networks
论文作者
论文摘要
我们检查了爱因斯坦田间方程的空间均匀和各向异性解的子集：Bianchi型A型模型，并表明它们可以写成连续的时间复发性神经网络（CTRNN）。爱因斯坦方程的这种重新制作使人们可以将潜在复杂的非线性方程式编写为更简单的动态系统，该系统由神经网络权重的线性组合和物流sigmoid激活函数组成。通过使用明确的runge-kutta求解器对CTRNN本身进行培训，以对Bianchi A型模型进行多个Einstein's方程的解决方案，然后使用非线性最小二乘方法来找到最佳的重量，时间延迟常数，以及为Ctrnnnnnneequrations方程提供最佳拟合。在数值示例方面，我们专门为Bianchi I型和II型模型提供了解决方案。我们在本文结束时，对最佳参数概率分布和未来工作的想法进行了一些评论。
We examine a subset of spatially homogenous and anisotropic solutions to Einstein's field equations: the Bianchi Type A models, and show that they can be written as a continuous-time recurrent neural network (CTRNN). This reformulation of Einstein's equations allows one to write potentially complicated nonlinear equations as a simpler dynamical system consisting of linear combinations of the neural network weights and logistic sigmoid activation functions. The CTRNN itself is trained by using an explicit Runge-Kutta solver to sample a number of solutions of Einstein's equations for the Bianchi Type A models and then using a nonlinear least-squares approach to find the optimal set of weights, time delay constants, and bias parameters that provide the best fit of the CTRNN equations to the Einstein equations. In terms of numerical examples, we specifically provide solutions to Bianchi Type I and II models. We conclude the paper with some comments on optimal parameter probability distributions and ideas for future work.
