论文标题
表征具有卷积神经网络的对数正态分数 - 布朗尼 - 动作密度场
Characterising lognormal fractional-Brownian-motion density fields with a Convolutional Neural Network
论文作者
论文摘要
在尝试从统计上量化星际介质的密度结构时，天文学家考虑了各种分形模型。在这里，我们认为，要正确地表征分形模型，需要精确定义用于生成密度字段的算法，并指定 - 至少 - 三个参数：一个参数约束字段的空间结构；一个参数限制了不同尺度上结构之间的密度对比。一个参数限制了预期自相似性的空间尺度的动态范围（由于物理考虑，或者是由于生成输入数据的观测技术或数值技术的局限性）。现实的分形领域也必须是嘈杂的和非周期性的。我们用指定的分数布朗运动（XFBM）算法来说明这一点，该算法很受欢迎，因为它提供了大约对数正常的密度字段，并且三个参数分别为power spectrum指数，$β$，$β$，指数因子，$ {\ cal s} $，以及动态范围，$ {$}。然后，我们探索并比较了可能用于估计这些参数的两种方法：机器学习和已建立的$δ$ - 变量程序。我们表明，价格为$ 2 \leqβ\ leq 4 $和$ 0 \ leq {\ cal s} \ leq 3 $，一个经过适当训练的卷积神经网络能够客观地估算$β$（带有root-mean-square错误$ $ $ $ε_ {_β} \ sim 0.12 $）和$ s}; $ε_ {_ {\ cal s}}} \ sim 0.29 $）。 $ \;δ$ - 变量也能够估计$β$，尽管错误的错误（$ε_{_β} \ sim 0.17 $），但进行了一些人为干预，但无法估计$ {\ cal s} $。
In attempting to quantify statistically the density structure of the interstellar medium, astronomers have considered a variety of fractal models. Here we argue that, to properly characterise a fractal model, one needs to define precisely the algorithm used to generate the density field, and to specify -- at least -- three parameters: one parameter constrains the spatial structure of the field; one parameter constrains the density contrast between structures on different scales; and one parameter constrains the dynamic range of spatial scales over which self-similarity is expected (either due to physical considerations, or due to the limitations of the observational or numerical technique generating the input data). A realistic fractal field must also be noisy and non-periodic. We illustrate this with the exponentiated fractional Brownian motion (xfBm) algorithm, which is popular because it delivers an approximately lognormal density field, and for which the three parameters are, respectively, the power spectrum exponent, $β$, the exponentiating factor, ${\cal S}$, and the dynamic range, ${\cal R}$. We then explore and compare two approaches that might be used to estimate these parameters: Machine Learning and the established $Δ$-Variance procedure. We show that for $2\leqβ\leq 4$ and $0\leq{\cal S}\leq 3$, a suitably trained Convolutional Neural Network is able to estimate objectively both $β$ (with root-mean-square error $ε_{_β}\sim 0.12$) and ${\cal S}$ (with $ε_{_{\cal S}}\sim 0.29$). $\;Δ$-variance is also able to estimate $β$, albeit with a somewhat larger error ($ε_{_β}\sim 0.17$) and with some human intervention, but is not able to estimate ${\cal S}$.
