<p align="center" > <img width="50%" src="/docs/aigle_logo.png" /> </p>

About AIGLETools

AIGLETools is a package written in Python, designed to minimize the effort required to build ab initio generalized Langevin equation (AIGLE) & ab initio Langevin equation (AILE) models for multi-dimensional time series data.

What is Generalized Langevin equation (GLE)

GLE is a non-Markovian equation of motion describing the time evolution of a system with generalized coordinates ${\mathbf{x}, \mathbf{p}}$:

$\dot{\mathbf{p}}(t) = -\nabla_{\mathbf{x}}G(\mathbf{x}(t)) + \int_0^t \mathbf{K}(s)\mathbf{p}(t-s) ds + \mathbf{R}(t) + \mathbf{\xi}(t);$

$\dot{\mathbf{x}}(t) = \mathbf{M}^{-1}\mathbf{p}(t)$.

For n-dimensional generalized position $\mathbf{x}$ and n-dimensional generalized momentum $\mathbf{p}$, $G(\mathbf{x})$ is the effective potential energy of the system. $\mathbf{\xi}(t)$ and $\mathbf{F}(t)$ are respectively the external driving force and the environmental noise. $\mathbf{K}(s)$ is a $n\times n$ memory kernel matrix, describing how the system at time $t$ responds to its historical state at time $t-s$. $M$ is a static $n\times n$ matrix, describing the inertia of the system.

What is Langevin equation (LE)

The Langevin equation(LE) is the Markovian limit of GLE, given as

$\dot{\mathbf{p}}(t) = -\nabla_{\mathbf{x}}G(\mathbf{x}(t)) - \eta \mathbf{p}(t) + \mathbf{w}(t) + \mathbf{\xi}(t);$

$\dot{\mathbf{x}}(t) = \mathbf{M}^{-1}\mathbf{p}(t)$.

Here, $\eta$ is a static $n\times n$ matrix, mimicking a friction acting on the system. $\mathbf{w}(t)$ is a white noise.

What is AIGLE and AILE?

AIGLE is the generalized Langevin equation extracted from the history of $\mathbf{x}(t)$ with few or no ad hoc assumptions. AILE is taken as the Markovian limit of AIGLE. Both AIGLE and AILE suppose to recover dynamical properties of $\mathbf{x}(t)$, while AIGLE is expected to be more faithful to the time series data than AILE.

Highlighted features

• Deal with high-dimensional and heterogeneous time-series data

• Integrate data processing, model training and simulation

  • User-friendly training of GLE model; Expertise in GLE not required.
  • Built-in multi-dimensional GLE/LE simulator
  • OPENMM plugin (Python interface)

• Exact enforcement of second fluctuation-dissipation theorem for long-term simulation

Credits

Please cite Xie, Pinchen, and Weinan E. "Coarse-Graining Conformational Dynamics with Multidimensional Generalized Langevin Equation: How, When, and Why." Journal of Chemical Theory and Computation (2024). (doi.org/10.1021/acs.jctc.4c00729) for general purposes.

Installation

pip install .

Use AIGLETools

The training and simulation of AIGLE/AILE model follow the steps below

<p align="center" > <img width="100%" src="/docs/workflow.png" /> </p>

/examples/harmonic_polymer demonstrates the whole workflow for particle-based coarse-graining. /examples/alaine_dipeptide demonstrates the whole workflow for collective variable-based coarse-graining.

Examples

Check /examples for Jupyter notebook demonstration of AIGLETools.

<!-- ## Units -->

Questions?

AIGLETools is under active development. More examples will be posted. If you have any question, send email to pinchenx at math dot princeton dot edu.