Dynamical system state estimation and parameter calibration problems are ubiquitous across science and engineering. Bayesian approaches to the problem are the gold standard as they allow for the quantification of uncertainties and enable the seamless fusion of different experimental modalities. When the dynamics are discrete and stochastic, one may employ powerful techniques such as Kalman, particle, or variational filters. Practitioners commonly apply these methods to continuous-time, deterministic dynamical systems after discretizing the dynamics and introducing fictitious transition probabilities. However, approaches based on time-discretization suffer from the curse of dimensionality since the number of random variables grows linearly with the number of time-steps. Furthermore, the introduction of fictitious transition probabilities is an unsatisfactory solution because it increases the number of model parameters and may lead to inference bias. To address these drawbacks, the objective of this paper is to develop a scalable Bayesian approach to state and parameter estimation suitable for continuous-time, deterministic dynamical systems. Our methodology builds upon information field theory. Specifically, we construct a physics-informed prior probability measure on the function space of system responses so that functions that satisfy the physics are more likely. This prior allows us to quantify model form errors. We connect the system's response to observations through a probabilistic model of the measurement process. The joint posterior over the system responses and all parameters is given by Bayes' rule. To approximate the intractable posterior, we develop a stochastic variational inference algorithm. In summary, the developed methodology offers a powerful framework for Bayesian estimation in dynamical systems.

翻译：动力学系统状态估计与参数标定问题普遍存在于科学与工程领域。贝叶斯方法因其能够量化不确定性并实现不同实验模态的无缝融合，被视为该领域的黄金标准。当系统动力学具有离散随机特性时，可采用卡尔曼滤波、粒子滤波或变分滤波等强大技术。实际应用中，研究者通常通过对连续时间确定性动力学系统进行离散化并引入虚拟转移概率来运用这些方法。然而，基于时间离散化的方法会面临维度灾难问题，因为随机变量数量随时间步长呈线性增长。此外，引入虚拟转移概率的解决方案存在固有缺陷：不仅增加了模型参数数量，还可能导致推断偏差。为克服这些不足，本文旨在开发一种适用于连续时间确定性动力学系统的可扩展贝叶斯状态与参数估计方法。我们的方法论建立在信息场理论基础上。具体而言，我们在系统响应的函数空间上构建了物理信息先验概率测度，使得满足物理规律的函数具有更高概率。该先验允许我们量化模型形式误差。我们通过测量过程的概率模型将系统响应与观测数据相关联。基于贝叶斯定理，可得到系统响应与所有参数的联合后验分布。为近似难以直接计算的后验分布，我们开发了随机变分推断算法。总之，本研究所提出的方法论为动力学系统贝叶斯估计提供了强有力的理论框架。