Solving differential equations with neural networks: Applications to the calculation of cosmological phase transitions
Solving differential equations with neural networks: Applications to the calculation of cosmological phase transitions
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DOI：
10.1103/physrevd.100.016002
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发表时间：
2019-07-09
影响因子：
5
通讯作者：
Waite, Philip
中科院分区：
文献类型：
--
作者：
Piscopo, Maria Laura;Spannowsky, Michael;Waite, Philip
从观察到人工神经网络是唯一适合解决优化问题，大多数物理问题可以作为一个优化任务，我们介绍了一种新的方法来寻找一个数值解广泛的微分方程类。我们发现我们的方法是非常灵活和稳定的，而不依赖于审判的解决方案，并适用于普通，偏微分方程和耦合。我们应用我们的方法来计算宇宙学相变的隧穿剖面，这是一个与重子生成和随机引力波谱相关的问题。将我们的解决方案与公开可用的代码进行比较，这些代码使用优化的数值方法来计算隧道剖面，我们发现我们的方法至少可以提供与这些专用微分方程求解器一样准确的结果，并且对于某些参数选择，甚至可以提供更准确和可靠的解决方案。特别是，我们比较了神经网络方法与两个公开的配置文件求解器，CosmoTransitions和BubbleProfiler，并给出明确的例子，神经网络方法找到正确的解决方案，而专用的求解器没有。我们指出，这种方法使用人工神经网络来解决方程是可行的任何问题，可以转换为形式F（（x）右箭头）= 0，因此适用于各种其他问题的微扰和非微扰量子场论。
Starting from the observation that artificial neural networks are uniquely suited to solving optimization problems, and most physics problems can be cast as an optimization task, we introduce a novel way of finding a numerical solution to wide classes of differential equations. We find our approach to be very flexible and stable without relying on trial solutions, and applicable to ordinary, partial and coupled differential equations. We apply our method to the calculation of tunneling profiles for cosmological phase transitions, which is a problem of relevance for baryogenesis and stochastic gravitational wave spectra. Comparing our solutions with publicly available codes which use numerical methods optimized for the calculation of tunneling profiles, we find our approach to provide at least as accurate results as these dedicated differential equation solvers, and for some parameter choices, even more accurate and reliable solutions. In particular, we compare the neural network approach with two publicly available profile solvers, CosmoTransitions and BubbleProfiler, and give explicit examples where the neural network approach finds the correct solution while dedicated solvers do not. We point out that this approach of using artificial neural networks to solve equations is viable for any problem that can be cast into the form F((x) over right arrow) = 0, and is thus applicable to various other problems in perturbative and nonperturbative quantum field theory.