# An information field theory approach to Bayesian state and parameter estimation in dynamical systems

arXiv: https://arxiv.org/abs/2306.02150 · HTML: https://mlsanity.com/paper/2306.02150/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 77 references that are in the map.
- First posted on arXiv 3 Jun 2023.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [Composable Effects for Flexible and Accelerated Probabilistic Programming in NumPyro](https://mlsanity.com/paper/1912.11554/) (2019)
- [Learning Non-Convergent Non-Persistent Short-Run MCMC Toward Energy-Based Model](https://mlsanity.com/paper/1904.09770/) (2019)
- [Learning Latent Space Energy-Based Prior Model](https://mlsanity.com/paper/2006.08205/) (2020)
- [Physics-informed neural network for ultrasound nondestructive quantification of surface breaking cracks](https://mlsanity.com/paper/2005.03596/) (2020)
- [Multi-fidelity Bayesian Neural Networks: Algorithms and Applications](https://mlsanity.com/paper/2012.13294/) (2020)
- [NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation framework](https://mlsanity.com/paper/2012.07938/) (2020)
- [Black-box Variational Inference for Stochastic Differential Equations](https://mlsanity.com/paper/1802.03335/) (2018)
- [Scalable Variational Inference for Dynamical Systems](https://mlsanity.com/paper/1705.07079/) (2017)
- [Physics-informed Information Field Theory for Modeling Physical Systems with Uncertainty Quantification](https://mlsanity.com/paper/2301.07609/) (2023)
- [Field dynamics inference for local and causal interactions](https://mlsanity.com/paper/1902.02624/) (2019)
- [Information field theory](https://mlsanity.com/paper/1301.2556/) (2013)

## Built on by (most cited first, 1 of 1)

- [An interpretation of the Brownian bridge as a physics-informed prior for the Poisson equation](https://mlsanity.com/paper/2503.00213/) (2025)

## Abstract (arXiv)

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.