标题： 信息场动力学的实际应用方面 显示英文标题

标题： On the Practical Applications of Information Field Dynamics

摘要： 在这项研究中，我们探索了偏微分方程的一种新的模拟方案，称为信息场动力学（IFD）。 信息场动力学试图通过将贝叶斯场推理纳入模拟方案中来改进现有的模拟方案。 场推理确实是贝叶斯的，因此依赖于先验信念的概念。 提出了一系列理论和实用的结果。 在探索IFD框架下可能存在的两类一般模拟方案之前，介绍了许多小修复和关于一般理论的结果。 对于两者，我们在开发原型方案的同时展示了一组理论结果。 这些模型中的第一类大致对应于传统的固定网格数值PDE求解器。 这些模型中的贝叶斯先验假设是场是平滑的，并且它们之间的相关结构不会因位置而变化。 出于这些原因，我们将它们称为平移不变方案。 我们展示了这些方案的稳定性要求，但最重要的是，我们证明了在高分辨率极限下，这些方案确实收敛到场的真实行为。 从未有任何人对任何以前的IFD方案证明过收敛性。 我们还找到了这些代码的误差标度，并表明它们实现了一些非常类似于高阶有限差分导数逼近的东西，这是最基础和最被研究的数值方案。 这是一个重要的结果，它证明了IFD方法的有效性。 第二类方案称为SPH类方案，类似于现有的平滑粒子流体力学代码，在这些代码中，模拟网格随着所模拟场的流动而移动。 显示英文摘要

摘要： In this study we explore a new simulation scheme for partial differential equations known as Information Field Dynamics (IFD). Information field dynamics attempts to improve on existing simulation schemes by incorporating Bayesian field inference into the simulation scheme. The field inference is truly Bayesian and thus depends on a notion of prior belief. A number of results are presented, both theoretical and practical. Many small fixes and results on the general theory are presented, before exploring two general classes of simulation schemes that are possible in the IFD framework. For both, we present a set of theoretical results alongside the development of a prototype scheme. The first class of models corresponds roughly to traditional fixed-grid numerical PDE solvers. The prior Bayesian assumption in these models is that the fields are smooth, and their correlation structure does not vary between locations. For these reasons we call them translation-invariant schemes. We show the requirements for stability of these schemes, but most importantly we prove that these schemes indeed converge to the true behaviour of the field in the limit of high resolutions. Convergence had never been shown for any previous IFD scheme. We also find the error scaling of these codes and show that they implement something very analogous to a high-order finite-difference derivative approximation, which are the most elementary and well-studied numerical schemes. This is an important result, which proves the validity of the IFD approach. The second class of schemes, called the SPH-like schemes are similar to existing Smooth Particle Hydrodynamics codes, in which the simulation grid moves with the flow of the field being modelled.