Feedback system identification, not necessarily linear

0.1 Basic Construction

There are a few variations. We start with the basic continuous time state space model.

Here we have an unobserved Markov state process $x(t)$ on $\mathcal{X}$ and an observation process $y(t)$ on $\mathcal{Y}$. For now they will be assumed to be finite dimensional vectors over $\mathbb{R}.$ They will additionally depend upon a vector of parameters $\theta$ We observe the process at discrete times $t(1:T)=(t_1,t_2,\dots ,t_T),$ and we write the observations $y(1:T)=(y(t_1),y(t_2),\dots ,y(1_T)).$

We presume our processes are completely specified by the following conditional densities (which might not have closed-form expression)

The transition density

$$f(x(t_i)|x(t_{i-1}),\theta )$$

The observation density…

TBC.

1 Method of adjoints

A trick in differentiation which happens to be useful in differentiating likelihood (or other functions) of time evolving systems using automatic differentiation. e.g. Errico (1997).

See the method of adjoints.

2 In particle filters

See particle filters for system identification.

3 Indirect inference

The simulator is a black box and we have access only to its inputs and outputs. Popular. See simulation-based inference.

4 Via variational filtering

I quite like this one egoistically because it connects to my own research. Dowling, Zhao, and Park (2024) catniplab/xfads: large scale variational Gaussian state-space models:

State-space graphical models and the variational autoencoder framework provide a principled apparatus for learning dynamical systems from data. State-of-the-art probabilistic approaches are often able to scale to large problems at the cost of flexibility of the variational posterior or expressivity of the dynamics model. However, those consolidations can be detrimental if the ultimate goal is to learn a generative model capable of explaining the spatiotemporal structure of the data and making accurate forecasts. We introduce a low-rank structured variational autoencoding framework for nonlinear Gaussian state-space graphical models capable of capturing dense covariance structures that are important for learning dynamical systems with predictive capabilities. Our inference algorithm exploits the covariance structures that arise naturally from sample based approximate Gaussian message passing and low-rank amortized posterior updates – effectively performing approximate variational smoothing with time complexity scaling linearly in the state dimensionality. In comparisons with other deep state-space model architectures our approach consistently demonstrates the ability to learn a more predictive generative model. Furthermore, when applied to neural physiological recordings, our approach is able to learn a dynamical system capable of forecasting population spiking and behavioural correlates from a small portion of single trials.

5 Learning SDEs

• Generalising Automatic Differentiation to Automatic Sparsity, Uncertainty, Stability, and Parallelism
• mitmath/18303 course
• How can the general Green’s function of a linear homogeneous differential equation be derived?
• ICERM SciML conf
• JB_grothendieck_proof.pdf
• kar12Supple.pdf
• Kolmogorov Backward Equations
• mitmath/18303: 18.303 - Linear PDEs course
• ModelingToolkit, Modelica, and Modia: The Composable Modeling Future in Julia
• Neural SDEs: Deep Generative Models in the Diffusion Limit - Maxim Raginsky
• Seminars & Workshops in DATA DRIVEN SCIENCE & ENGINEERING

6 Tooling

• AaltoML/SDE: Example codes for the book Applied Stochastic Differential Equations
• An Implicit/Explicit CUDA-Accelerated Solver for the 2D Beeler-Reuter Model
• DiffEqFlux.jl – A Julia Library for Neural Differential Equations
• DiffEqFlux.jl: Generalized Physics-Informed and Scientific Machine Learning (SciML) · DiffEqFlux.jl
• DifferentialEquations.jl: Scientific Machine Learning (SciML) Enabled Simulation and Estimation · DifferentialEquations.jl
• Diffrax
• dnncode/Spatio-Temporal-Model-for-SPDE: Statistical Modeling for Spatio-Temporal Data from Physical Convection-Diffusion Processes
• JuliaSim
• NeuralNetDiffEq.jl: A Neural Network solver for ODEs
• Noise Processes · DifferentialEquations.jl
• Parameter Estimation and Bayesian Analysis · DifferentialEquations.jl
• Physics-Informed Neural Networks solver · NeuralPDE.jl
• SciML: Open Source Software for Scientific Machine Learning
• SciML/DiffEqFlux.jl: Universal neural differential equations with O(1) backprop, GPUs, and stiff+non-stiff DE solvers, demonstrating scientific machine learning (SciML) and physics-informed machine learning methods
• SciML/NeuralPDE.jl: Physics-Informed Neural Networks (PINN) and Deep BSDE Solvers of Differential Equations for Scientific Machine Learning (SciML) accelerated simulation
• SciML/SciMLTutorials.jl: Tutorials for doing scientific machine learning (SciML) and high-performance differential equation solving with open source software.
• Solving Systems of Stochastic PDEs and using GPUs in Julia
• willtebbutt/TemporalGPs.jl: Fast inference for Gaussian processes in problems involving time

7 Incoming

• Corenflos et al. (2021) describe an optimal transport method
• Campbell et al. (2021) describes variational inference that factors out the unknown parameters.
• Gu et al. (2021) unifies neural ODEs with RNNs.

8 References