Differential equation models are used in a wide variety of scientific fields to describe the behaviour of physical systems. Commonly, solutions to given systems of differential equations are not available in closed-form; in such situations, the solution to the system is generally approximated numerically. The numerical solution obtained will be systematically different from the (unknown) true solution implicitly defined by the differential equations. Even if it were known, this true solution would be an imperfect representation of the behaviour of the real physical system that it was designed to represent. A Bayesian framework is proposed which handles all sources of numerical and structural uncertainty encountered when using ordinary differential equation (ODE) models to represent real-world processes. The model is represented graphically, and the graph proves to be useful tool, both for deriving a full prior belief specification and for inferring model components given observations of the real system. A general strategy for modelling the numerical discrepancy induced through choice of a particular solver is outlined, in which the variability of the numerical discrepancy is fixed to be proportional to the length of the solver time-step and a grid-refinement strategy is used to study its structure in detail. A Bayes linear adjustment procedure is presented, which uses a junction tree derived from the originally specified directed graphical model to propagate information efficiently between model components, lessening the computational demands associated with the inference. The proposed framework is illustrated through application to two examples: a model for the trajectory of an airborne projectile moving subject to gravity and air resistance, and a model for the coupled motion of a set of ringing bells and the tower which houses them.

微分方程模型在广泛的科学领域中用于描述物理系统的行为。通常，对于给定的微分方程组，其解决方案不是封闭形式的。在这种情况下，系统的解通常在数值上近似。所获得的数值解将与由微分方程隐式定义的（未知）真解在系统上有所不同。即使已知，这种真正的解决方案也不能完美地表示其设计要代表的真实物理系统的行为。提出了一种贝叶斯框架，该框架可以处理使用常微分方程（ODE）模型表示实际过程时遇到的所有数值和结构不确定性源。该模型以图形方式表示，该图被证明是有用的工具，既可以用于得出完整的先验置信规范，又可以在给定对真实系统的观察的情况下推断模型组件。概述了对通过选择特定求解器而引起的数值差异进行建模的一般策略，其中，数值差异的可变性固定为与求解器时间步长成比例，并且使用网格细化策略进行研究其结构详细。提出了一种贝叶斯线性调整程序，该程序使用从最初指定的有向图形模型派生的结点树在模型组件之间有效地传播信息，从而减少了与推理相关的计算需求。通过应用到两个示例来说明所建议的框架：