We introduce neural information field filter, a Bayesian state and parameter estimation method for high-dimensional nonlinear dynamical systems given large measurement datasets. Solving such a problem using traditional methods, such as Kalman and particle filters, is computationally expensive. Information field theory is a Bayesian approach that can efficiently reconstruct dynamical model state paths and calibrate model parameters from noisy measurement data. To apply the method, we parameterize the time evolution state path using the span of a finite linear basis. The existing method has to reparameterize the state path by initial states to satisfy the initial condition. Designing an expressive yet simple linear basis before knowing the true state path is crucial for inference accuracy but challenging. Moreover, reparameterizing the state path using the initial state is easy to perform for a linear basis, but is nontrivial for more complex and expressive function parameterizations, such as neural networks. The objective of this paper is to simplify and enrich the class of state path parameterizations using neural networks for the information field theory approach. To this end, we propose a generalized physics-informed conditional prior using an auxiliary initial state. We show the existing reparameterization is a special case. We parameterize the state path using a residual neural network that consists of a linear basis function and a Fourier encoding fully connected neural network residual function. The residual function aims to correct the error of the linear basis function. To sample from the intractable posterior distribution, we develop an optimization algorithm, nested stochastic variational inference, and a sampling algorithm, nested preconditioned stochastic gradient Langevin dynamics. A series of numerical and experimental examples verify and validate the proposed method.

翻译：我们提出神经信息场滤波器，这是一种针对高维非线性动力系统在大规模测量数据集下的贝叶斯状态与参数估计方法。使用传统方法（如卡尔曼滤波器与粒子滤波器）求解此类问题计算成本高昂。信息场理论是一种贝叶斯方法，能够从含噪声测量数据中高效重建动力模型状态路径并校准模型参数。为应用该方法，我们采用有限线性基的跨度对时间演化状态路径进行参数化。现有方法需通过初始状态对状态路径进行重新参数化以满足初始条件。在未知真实状态路径的情况下，设计表达能力强且简洁的线性基对推断精度至关重要，但具有挑战性。此外，基于初始状态对状态路径进行重新参数化在线性基中易于实现，但对于更复杂、表达能力更强的函数参数化（如神经网络）则非易事。本文目标是为信息场理论方法简化和丰富使用神经网络的状态路径参数化类别。为此，我们提出一种使用辅助初始状态的广义物理信息条件先验。我们证明现有重新参数化方法是其特例。我们采用残差神经网络对状态路径进行参数化，该网络由线性基函数与傅里叶编码全连接神经网络残差函数构成。残差函数旨在修正线性基函数的误差。为从难以处理的後验分布中采样，我们开发了优化算法——嵌套随机变分推断，以及采样算法——嵌套预处理随机梯度朗之万动力学。一系列数值与实验案例验证了所提方法的有效性。