Mechanistic models with differential equations are a key component of scientific applications of machine learning. Inference in such models is usually computationally demanding, because it involves repeatedly solving the differential equation. The main problem here is that the numerical solver is hard to combine with standard inference techniques. Recent work in probabilistic numerics has developed a new class of solvers for ordinary differential equations (ODEs) that phrase the solution process directly in terms of Bayesian filtering. We here show that this allows such methods to be combined very directly, with conceptual and numerical ease, with latent force models in the ODE itself. It then becomes possible to perform approximate Bayesian inference on the latent force as well as the ODE solution in a single, linear complexity pass of an extended Kalman filter / smoother - that is, at the cost of computing a single ODE solution. We demonstrate the expressiveness and performance of the algorithm by training, among others, a non-parametric SIRD model on data from the COVID-19 outbreak.

翻译：具有不同方程式的机械模型是机器学习科学应用的一个关键组成部分。 在这类模型中,推论通常具有计算要求,因为它涉及反复解决差异方程式。这里的主要问题是数字求解器很难与标准推算技术相结合。最近关于概率数字模型的工作为普通差异方程式开发了一个新的解答器类别,直接用贝叶斯过滤法来描述解析过程。我们在这里表明,这允许这些方法在概念和数字上非常直接地结合,在概念和数字上容易,与ODE本身的潜在力模型相结合。然后有可能在一个扩展的卡尔曼过滤器/光滑器的单一、线性复杂通道上对潜力和ODE解决方案进行近似贝叶斯式推论,也就是说,以计算单一的解算法解决方案为代价。我们通过培训一个非参数的SIRD模型,从COVID-19爆发的数据中演示算算算算算算算算算算算法的表达性和性。