Nonparametric Differential Equations (npde) with Gaussian Processes

This repository contains a Python implementation of npde - a nonparametric model for learning unknown differential equations. Two related papers are

• Nonparametric stochastic differential equations
• Nonparametric ordinary differential equations

Also, this repository overrides the old MATLAB implementation of our ODE model, which was published along with the paper.

Demo Notebook

More details, figures and usage examples can be found in this demo notebook.

Python Code

The implementation is in Python3.5, and it requires TensorFlow(1.6.0+) and GPflow(1.1+). The dependencies can be installed via

# in a python3.5 installation
pip install -r requirements.txt

Fitting our model on some data is as simple as follows:

npde = build_model(sess, t, Y, model='ode')
npde = fit_model(sess, npde, t, Y) # parameters are the TensorFlow session, time points and observations. model could be ode/sde

It is also possible to predict the future

path = npde.predict(x0,t) # x0 and t are the time points and the initial value

and sample from the learned model

samples = npde.sample(x0,t,Nw) # Nw is the number of samples

demo.ipynb contains some more details, including three example fits where the underlying dynamics are governed by an ODE, SDE with constant diffusion, and SDE with state dependent diffusion. Below are ODE and constant diffusion fits:

Example ODE Fit

Example SDE Fit

SDE Samples

The wrapper function build_model is where the parameters, kernel, and posterior are initialized - perhaps a good starting point to read the code. Check out the following optional inputs to better understand what's what:

• sf: drift signal variance
• ell: drift lengthscale
• sfg: diffusion signal variance
• ellg: diffusion lengthscale
• W: inducing point grid width
• Nw: number of random draws used for optimizing SDEs