Neural Lyapunov Functions for Nonlinear Systems

DeepDynaSim collection | Learning and stability analysis | TensorFlow / MATLAB

Explore deep neural-network approximations of Lyapunov functions and their orbital derivatives. This research and teaching repository contains two Python notebooks, MATLAB stability examples and a technical report.

Repository map

Resource	Purpose
Lyapunov_Approximation_1.ipynb	Neural approximation example for a two-dimensional state
Lyapunov_Approximation_2.ipynb	Neural approximation example for a four-dimensional state
evaluateStability.m	MATLAB stability-condition evaluation helper
example_stability.m	Illustrative MATLAB workflow with placeholder training data
Technical report	Method background and discussion

Getting started

The notebooks import TensorFlow, NumPy and Matplotlib. Open either notebook in a compatible Jupyter environment and review the system dynamics, sampling region, training objective and plotting cells before running them in order. Dependency versions are not pinned in this repository.

git clone https://github.com/DeepDynaSim/DeepLyapunovFunction.git
cd DeepLyapunovFunction
python -m venv .venv
# Activate the environment using the command for your operating system.
python -m pip install jupyter tensorflow numpy matplotlib
python -m notebook

For MATLAB, inspect example_stability.m and evaluateStability.m together. Neural-network functions in the example require the corresponding MATLAB toolbox. The random training targets and other placeholders must be adapted to the intended dynamical system.

Interpreting the examples

A fitted scalar function or a sampled numerical check does not by itself establish a formal stability certificate over a continuous domain. Examine positivity, orbital-derivative conditions, the domain and the verification method for the specific system. The examples support method exploration and visual analysis.

Known limitations of the historical examples

• The scalar and batched dynamics in Lyapunov_Approximation_1.ipynb differ: one cell uses x1_dot=-x1^3, while the batched cell uses x1_dot=-x1-10*x2^2. They must not be interpreted as evaluations of the same vector field without reconciling that choice.
• The four-state notebook includes x1_dot=-x1^2. For a negative initial state, x1(t)=x1(0)/(1+x1(0)*t) has finite escape; x1(0)=-0.5 is singular at t=2. Thus this example does not establish two-sided asymptotic stability of the origin. A low value-regression loss does not repair that problem.
• evaluateStability.m differentiates V symbolically and therefore requires Symbolic Math Toolbox and a candidate that accepts symbolic inputs. An arbitrary trained network handle may not support this operation. Its Boolean result describes only the supplied finite sample set.
• The first example in example_stability.m is a rotation: its quadratic candidate has zero orbital derivative and is expected to fail the requested strict negative-decay test. That failure does not prove the rotation is Lyapunov unstable.

The MATLAB example now forms a full Cartesian grid rather than pairing identical coordinate lists. For an independent, strictly decreasing numerical example:

axisValues = -1:0.5:1;
[X1, X2] = meshgrid(axisValues, axisValues);
domain = [X1(:)'; X2(:)'];
passedSamples = evaluateStability(@(x) -x, ...
    @(x) x(1)^2+x(2)^2, @(x) 2*sum(x.^2), domain);

Passing those samples is not a continuum certificate. The separate structure-preserving neural Lyapunov repository contains explicit proof assumptions and a distinction between sampled evidence and validated verification.

The current checkout does not include export_to_onnx.py or a complete Simulink integration workflow. ONNX export and deployment require an additional implementation.

Contributing and citation

For a useful issue or contribution, include the notebook/file, environment versions, system equations, sampling settings and a minimal reproducible example. Cite the repository and the commit/version used in academic work. The existing GPL-2.0 license governs reuse.

Author contact retained from the original repository: Caglar Uyulan, caglaruyulan1@gmail.com.

Related work and collaboration

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DeepDynaSim is an engineering R&D and consulting company in İzmir, Türkiye, focused on physics-guided AI, simulation and control. For engineering collaboration: support@deepdynasim.com. For repository questions, use Issues.

Reuse follows the existing LICENSE; academic authorship and third-party attribution remain with their respective creators.